Many real-world multi-objective optimization problems exhibit L-shaped Pareto fronts, characterized by steep trade-offs between objectives near their extreme values. This class of problems poses significant challenges for evolutionary algorithms in obtaining uniformly spread solutions across the Pareto front (PF). This paper introduces eight new test functions with L-shaped and reflected L-shaped PFs, intended to provide a valuable framework for benchmarking multi-objective optimization algorithms. The functions are based on modifications of the well-known DTLZ2 problem and a reciprocal function formulation, each formulated in standard and 'hard' variants. The 'hard' variants introduce modifications to the auxiliary functions, increasing problem difficulty by biasing the distribution of non-Pareto solutions away from the PF as the problem dimensionality increases. Numerical experiments are conducted using NSGA-II and MOEA/D algorithms, with performance evaluated using hypervolume and inverted generational distance metrics. The results demonstrate that the proposed test functions effectively challenge the algorithms, especially in their 'hard' variants, and outline the differences in algorithm performance based on the shape of the PF. These findings highlight the need for development of more robust multi-objective optimization techniques capable of handling such PF geometries.

Given a bilevel optimization problem, it is often implicitly assumed that all lower level variables participate in the lower level objective/constraint evaluations. However, in theory, there can be scenarios where this assumed variable-to-objective relation is misleading. This means that some variables purportedly associated with lower level objectives might not actually contribute to their calculation. Some lower level variables may only participate in upper level objective calculation, making hard to determine their desired values through the lower-level optimization process. When such solutions are passed to the upper level for evaluation, the upper level search can easily be misled by these inferior values. This property of certain problems poses a significant challenge to the solution methodologies, and this aspect has been generally overlooked in the literature. In this paper, we propose a simple pre-processing method aimed at identifying the association between variables and objectives through a causal relation test. To ascertain if an LL variable contributes to UL/LL objective(s), it is perturbed from a baseline value, and the resulting impact on the objective(s) is observed to make the inference. Then, an improved search scheme is formulated to allocate the evaluations effectively based on the identified variable-to-objective relations. Empirical experiments demonstrate that the proposed algorithm exhibits competitive performance across various problem settings, and are particularly beneficial for so-called deceptive bilevel problems.

Automated machine learning (AutoML) has emerged as a pivotal tool for applying machine learning (ML) models to real-world problems. Tree-based pipeline optimization tool (TPOT) is an AutoML framework known for effectively solving complex tasks. TPOT's search involves two fundamental objectives: finding optimal pipeline structures (i.e., combinations of ML operators) and identifying suitable hyperparameters for these structures. While its use of genetic programming enables TPOT to excel in structural search, its hyperparameter search, involving discretization and random selection from extensive potential value ranges, can be computationally inefficient. This paper presents a novel methodology that heavily restricts the initial hyperparameter search space, directing TPOT's focus towards structural exploration. As the search evolves, Bayesian optimization (BO) is used to refine the hyperparameter space based on data from previous pipeline evaluations. This method leads to a more targeted search, crucial in situations with limited computational resources. Two variants of this approach are proposed and compared with standard TPOT across six datasets, with up to 20 features and 20,000 samples. The results show the proposed method is competitive with canonical TPOT, and outperforms it in some cases. The study also provides new insights into the dynamics of pipeline structure and hyperparameter search within TPOT.

A number of real-world problems involve optimization of multiple conflicting criteria assessed using expensive evaluations. In practice, such expensive multi/many-objective optimization problems (EMOPs) need to be solved using small number of design evaluations. Surrogate-assisted evolutionary algorithms (SAEAs) are commonly used to solve EMOPs and broadly follow either a generational or a steady-state form, both having their characteristic advantages. The generational forms involve evaluation of multiple solutions in each generation, and are more suitable for scenarios where the parallelization of the evaluations is possible. The steady-state forms evaluate one solution at a time, therefore incorporating the additional information more frequently, but are more suited to non-parallelizable scenarios. Although it is possible to run the generational algorithms in a steady-state manner through certain parameter settings, we show in this study that such settings have a negative impact on their search performance. This leads to an inference that existing methods are not able to seamlessly switch between generational and steady-state forms based on the application at hand. In this study, we address this gap by extending a recently proposed steady-state framework (SASSEA) to a generalized form (GSAEA) which can be run in both steady-state or generational form by simply prescribing the number of true evaluations per generation. Numerical experiments are conducted using low evaluation budgets on a range of MOPs with up to 7 objectives and diverse types of Pareto-optimal fronts. The results indicate that the proposed framework shows competitive or better performance relative to the state-of-the-art methods, both in generational and steady-state setting.

Bilevel optimization problem (BLOP) refers to a class of problems with a hierarchical structure, wherein a lower level optimization problem acts as a constraint for an upper level optimization problem. Evolutionary algorithms (EAs) have been commonly used to solve BLOPs where underlying functions are black-box or do not conform to certain mathematical properties. One of the downsides of using EAs, especially in a nested format, is the significant computational expense (number of function evaluations); and a number of strategies have been proposed to mitigate this. However, most of the existing studies in the domain of evolutionary bilevel optimization are directed towards problems with single-objective at both levels, while very few have explored BLOPs with multiple objectives at one or both levels (BLMOPs). In this study, we investigate the potential benefits of utilizing knowledge transfer by seeding initial population from neighboring solutions for solving BLMOPs. Towards this end, we construct two simple strategies, referred to as full population transfer and selective population transfer, and study their potential to improve the performance over the baseline nested EA for BLMOPs. Experimental results show that the selective transfer strategy has more reliable and competitive performance compared to baseline. Empirical analysis is presented to highlight the relevant factors that lead to the observed performance trends.

Tree-based pipeline optimization tool (TPOT) is used to automatically construct and optimize machine learning pipelines for classification or regression tasks. The pipelines are represented as trees comprising multiple data transformation and machine learning operators --- each using discrete hyper-parameter spaces --- and optimized with genetic programming. During the evolution process, TPOT evaluates numerous pipelines which can be challenging when computing budget is limited. In this study, we integrate TPOT with Bayesian Optimization (BO) to extend its ability to search across continuous hyper-parameter spaces, and attempt to improve its performance when there is a limited computational budget. Multiple hybrid variants are proposed and systematically evaluated, including (a) sequential/periodic use of BO and (b) use of discrete/continuous search spaces for BO. The performance of these variants is assessed using 6 data sets with up to 20 features and 20,000 samples. Furthermore, an adaptive variant was designed where the choice of whether to apply TPOT or BO is made automatically in each generation. While the variants did not produce results that are significantly better than "standard" TPOT, the study uncovered important insights into the behavior and limitations of TPOT itself which is valuable in designing improved variants.

Most of the existing wind power comes from the use of traditional Horizontal-Axis Wind Turbines (HAWT), which typically require installation over vast countryside areas (in wind farms) due to considerations such as wake effects and noise. Recently, there has been an increasing interest to explore an alternative solution - Vertical-Axis Wind Turbines (VAWT) - that may be more suitable for compact urban areas. In this paper, we conduct design optimization of twin-blade VAWTs by evaluating the candidate designs through direct small-scale prototyping and physical experiments in-loop. The problem is a practical example of expensive optimization where the number of evaluations affordable are severely limited. In addition to the conventional single-objective form (maximize rotational speed), we also solve a multi-objective version of the problem (maximize rotational speed and minimize mass). For conducting optimization, we leverage surrogate-assisted approaches that make use of both predicted mean and uncertainties in modeling for an efficient exploration of the design space. The experiments demonstrate that the approach is able to generate non-intuitive designs that are competitive or better than the baseline (classic twin-blade Savonius design) within a small evaluation budget. The study also strengthens the case for applying this approach for design optimization in general.

Community microgrids, as an emerging technology, offer resiliency in operation for smart grids. Microgrids are seeing an increased penetration of eco-friendly electric vehicles (EVs) in recent years. However, the uncontrolled charging of EVs can easily overwhelm such electric networks. In this work, we propose an efficient demand response (DR) scheme based on dynamic pricing to enhance the capacity of the microgrid to securely host a large number of EVs. A hierarchical two-level optimization framework is introduced to realize the DR scheme. At the upper level, the dynamic prices for the participating users in DR are optimized while at the lower level, each user optimizes its energy consumption based on the price signal from the upper level. An evolutionary algorithm and a mixed-integer linear programming model is employed to solve the upper and lower level problems, respectively. Energy scheduling problems of the users are solved in a distributed manner which adds to the scalability of the approach. The proposed DR scheme is tested on a microgrid system adopted from the IEEE European low-voltage distribution network. Numerical experiments confirm the effectiveness of the proposed DR scheme compared to the benchmark pricing policies from the literature.

Engineering design optimization often involves use of numerical simulations to assess the performance of candidate designs. The simulations for computing high-fidelity (HF) performance estimates, such as finite element analysis or computational fluid dynamics, are typically computationally expensive. In some cases, it may also be possible to run an alternate or cheaper version of the simulation (through, e.g., use of a coarse mesh) to yield a low-fidelity (LF) performance estimate. Multifidelity optimization refers to the class of methods that aim to manage LF and HF evaluations efficiently to optimize computationally expensive problems within a limited computing budget. Among the prominent existing multifidelity approaches, some of them depend on a sufficiently dense a priori sampling; while others use unidirectional information exchange from LF to HF; both of which lead to a possibility of spending evaluation budget on unpromising search regions. This article proposes an improved multifidelity approach using an iterative, two-stage scheme (MFITS). It uses the collective information from the previously evaluated designs to determine a sampling neighborhood for LF evaluations. These samples are, in turn, used for building a co-kriging surrogate model that is then searched globally to identify a good candidate for HF evaluation. By restricting the LF sampling neighborhood, the computational budget can be used more efficiently, as the search is focused on regions that have historically produced good quality solutions. Numerical experiments and benchmarking are conducted on two suites of test problems and two practical design optimization problems to demonstrate the efficacy of MFITS.

Expensive multiobjective optimization problems (EMOPs) refer to those wherein evaluation of each candidate solution incurs a significant cost. To solve such problems within a limited number of solution evaluations, surrogate-assisted evolutionary algorithms (SAEAs) are often used. However, existing SAEAs typically operate in a generational framework wherein multiple solutions are identified for evaluation in each generation. There exist relatively few proposals in steady-state framework, wherein only a single solution is evaluated in each iteration. The development of such algorithms is crucial to efficiently solve EMOPs for which the evaluation of candidate designs cannot be parallelized. Furthermore, regardless of the framework used, the performance of current SAEAs tends to degrade when the Pareto front (PF) of the problem has irregularities, such as extremely concave/convex segments, even for 2/3-objective problems. To contextualize the motivation of this study, the performance of a few state-of-the-art SAEAs is first demonstrated on some such selected problems. Then, to address the above research gaps, we propose a surrogate-assisted steady-state EA (SASSEA), which incorporates a number of novel elements, including: 1) effective use of model uncertainty information to aid the search, including the use of the probabilistic dominance and Mahalanobis distance; 2) two-step infill identification using nondominance (ND) and distance-based selection; and 3) a shadow ND mechanism to avoid repeated selection and evaluation of dominated solutions. The efficacy of the proposed approach is demonstrated through extensive benchmarking on a range of test problems. It shows competitive performance relative to many state-of-the-art SAEAs, including both steady-state and generational approaches.

In solving multi-modal, multi-objective optimization problems (MMOPs), the objective is not only to find a good representation of the Pareto-optimal front (PF) in the objective space but also to find all equivalent Pareto-optimal subsets (PSS) in the variable space. Such problems are practically relevant when a decision maker (DM) is interested in identifying alternative designs with similar performance. There has been significant research interest in recent years to develop efficient algorithms to deal with MMOPs. However, the existing algorithms still require prohibitive number of function evaluations (often in several thousands) to deal with problems involving as low as two objectives and two variables. The algorithms are typically embedded with sophisticated, customized mechanisms that require additional parameters to manage the diversity and convergence in the variable and the objective spaces. In this letter, we introduce a steady-state evolutionary algorithm for solving MMOPs, with a simple design and no additional userdefined parameters that need tuning compared to a standard EA. We report its performance on 21 MMOPs from various test suites that are widely used for benchmarking using a low computational budget of 1000 function evaluations. The performance of the proposed algorithm is compared with six state-of-the-art algorithms (MO Ring PSO SCD, DN-NSGAII, TriMOEA-TA&R, CPDEA, MMOEA/DC and MMEA-WI). The proposed algorithm exhibits significantly better performance than the above algorithms based on the established metrics including IGDX, PSP and IGD. We hope this study would encourage design of simple, efficient and generalized algorithms to improve its uptake for practical applications.

Bilevel optimization refers to a challenging class of problems where a lower level (LL) optimization task acts as a constraint for an upper level (UL) optimization task. When a bilevel problem is solved using a nested evolutionary algorithm (EA), a large number of function evaluations are consumed since an LL optimization needs to be conducted to evaluate every candidate UL solution. Knowledge transfer of optimal LL solutions between neighboring UL solutions is a plausible approach to improve the search efficiency. Even though some of the past studies have utilized this strategy intuitively, the specific impact of the transferred solution(s) has not been clearly differentiated since it forms only a small component of a much more elaborate search framework. In this study, we intend to examine closely the effectiveness of direct solution transfer. To do so, the transferred solution (LL optimum of the nearest UL solution) is considered as the mainstay of the LL search, acting as the starting point for a direct local LL search. We first observe the performance of this approach on existing benchmarks. Based on the understanding gained from the experiments, we design modified problems where such a direct transfer is likely to face significant challenges. We then propose an improved approach that uses solution transfer more selectively by considering correlations between neighboring landscapes for a more effective transfer. Numerical experiments are conducted to demonstrate the challenges faced by the direct transfer on the modified problems, as well as the competitive performance of the correlation-based approach. We hope that the insights gained from the study will be beneficial for future development of efficient transfer-based approaches for bilevel optimization.

Bilevel optimization refers to a specialized class of problems where the optimum of an upper level (UL) problem is sought subject to the optimality of a nested lower level (LL) problem as a constraint. This nested structure necessitates a large number of function evaluations for the solution methods, especially population-based metaheuristics such as evolutionary algorithms (EAs). Reducing this effort remains critical for practical uptake of bilevel EAs, particularly forcomputationally expensiveproblems where each solution evaluation may involve a significant cost. This letter aims to contribute toward this field by a novel and previously unexplored proposition that bilevel optimization problems can be posed asmultifidelityoptimization problems. The underpinning idea is that an informed judgment of how accurate the LL optimum estimate should be to confidently determine its ranking can significantly cut down redundant evaluations during the search. Toward this end, we propose an algorithm which learns the appropriate fidelity to evaluate a solution during the search based on the seen data, instead of resorting to an exhaustive LL optimization. Numerical experiments are conducted on a range of standard as well as more complex variants of the SMD test problems to demonstrate the advantages of the proposed approach when compared to state-of-the-art surrogate-assisted algorithms.

Bilevel optimization refers to a specialized class of problems where one optimization task is nested as a constraint within another. Such problems emerge in a range of real-world scenarios involving hierarchical decision-making, and significant literature exists on classical and evolutionary approaches to solve them. However, computationally expensive bilevel optimization problems remain relatively less explored. Since each evaluation incurs a significant computational cost, one can only perform a limited number of function evaluations during the course of search. Surrogate-assisted strategies provide a promising way forward to deal with such classes of problems. Of particular interest to this study are the steady-state strategies which carefully pre-select a promising solution for true evaluation based on a surrogate model. The main aim of this paper is to compare two widely adopted steady-state infill strategies -Kriging believer (KB) and expected improvement (EI) - through systematic experiments within a nested optimization framework. Our experiments on a set of benchmark problems reveal some interesting and counter-intuitive observations. We discuss some of the underlying reasons and believe that the findings will inform further research on understanding and improving search strategies for expensive bilevel optimization.

Evolutionary multi-objective optimization (EMO) is often used to deal with practical problems in engineering design to identify products that exhibit the best possible compromise between multiple conflicting performance criteria. Much of the literature in EMO considers algorithmic developments and benchmarking problems involving a single concept only. However, in practice, there could be many applications where the solution of a given problem may be represented using multiple concepts, and optimizing each one of them individually to obtain the overall Pareto front may be inefficient. To address this gap, in this study, we firstly develop computer-aided models of multiple concepts for a simulation-based problem which can serve as a good application of multi-concept optimization. The problem involves the design of lattice structures with different types of unit cells (each constituting a concept), which can be customized to suit a range of real-world applications such as design of structures with minimal thermal expansion, low weight and high rigidity etc. Furthermore, we develop baseline evolutionary strategies to search across multiple concepts simultaneously. Empirical experiments show that the presented strategies are able to outperform conventional single concept-based approach. Moreover, some challenges are also uncovered which prompts the need for further developments.

Constrained optimization problems (COPs) are frequently encountered in real-world design applications. For some COPs, the evaluation of the objective(s) and/or constraint(s) may involve significant computational/temporal/financial cost. Such problems are referred to asexpensiveCOPs (ECOPs). Surrogate modeling has been widely used in conjunction with optimization methods for such problems, wherein the search is partially driven by an approximate function instead of true expensive evaluations. However, for any true evaluation, nearly all existing methods compute all objective and constraint values together as one batch. Suchfull evaluationapproaches may be inefficient for cases where the objective/constraint(s) can be evaluated independently of each other. In this article, we propose and study a constraint handling strategy for ECOPs usingpartialevaluations. The constraints are evaluated in a sequence determined based on their likelihood of being violated; and the evaluation is aborted if a constraint violation is encountered. Modified ranking strategies are introduced to effectively rank the solutions using the limited information thus obtained, while saving on significant function evaluations. The proposed algorithm is compared with a number of its variants to establish the utility of its key components systematically. Numerical experiments and benchmarking are conducted on a range of mathematical and engineering design problems to demonstrate the efficacy of the approach compared to state-of-the-art evolutionary optimization approaches.

In practical multi/many-objective optimization problems, a decision maker is often only interested in a handful of solutions of interest (SOI) instead of the entire Pareto Front (PF). It is therefore of significant research interest to design algorithms that can automatically detect SOIs and search around them instead of attempting to find the entire PF. However, this is challenging for a number of reasons. First and foremost, the interpretation of the underlying measures in terms of quantifying trade-off information for SOIs is not straightforward. Scalability is also an issue for most of such existing measures. Additionally, for many-objective algorithms that rely on decomposition, adaptation of reference directions and appropriate means to scale the objectives to maintain solution density around SOIs is not trivial. Lastly, constraints and decision-space are often overlooked in the existing studies but are important for practical applications. In this work, we present a simple approach to identify SOIs, using normalized net gain over nadir point and angle of influence. We illustrate the utility of the measure for offline and online identification of SOIs using a range of unconstrained and constrained benchmarks and practical design problems spanning up to 5 objectives. We also show further analysis in decision-space for an application problem to aid decision-making in practical scenarios.

This study investigates the makespan minimization problem that arises when compiling a general class of quantum algorithms into near-term quantum hardware. The problem is referred to as quantum circuit compilation problem (QCCP). Two new heuristics are proposed for solving the problem. Firstly, a rollout based sequential decision making approach is investigated. This heuristic decides on which operation to schedule next based on the makespan projection given by a guiding priority rule. Secondly, a stochastic version of rollout heuristic is proposed which iteratively switches between rollout and a simple priority rule to explore the search space. The two proposed heuristics are able to show improvement in makespan across a range of instance sizes and characteristics.

Real-world optimization problems involve constraints that must be satisfied for the design to be viable. The constraints are a manifestation of statutory physical limitations such as allowable strength, geometric compatibility or other practical considerations such as cost and time required for manufacturing. Constraint handling is thus an important area in the domain of optimization and there exists rich literature on the subject. Within population based stochastic optimization methods, constraint handling is typically implemented through a ranking process, where feasible solutions are considered better then infeasible ones. Recent studies have suggested that preserving infeasible solutions can be advantageous to the evolutionary search. Such studies have typically considered a paradigm where all objectives and constraints are evaluated simultaneously. In many practical scenarios however, it is possible to evaluate them independently. This opens up opportunities to gain computational benefits through sequencing constraint evaluation and using partial evaluation (i.e. only evaluate some of the constraints). In this paper, we systematically construct and study the performance of these strategies by combining them in different ways (total 8 variants). The numerical experiments compare the performance of these strategies under different evaluation and ranking scenarios. The study offers understanding of the advantages that can be gained by using appropriate combinations of these strategies for the cases where objective and constraint(s) are associated with individual cost and can be computed independently.

Many-objective optimization problems (MaOPs) contain four or more conflicting objectives to be optimized. A number of efficient decomposition-based evolutionary algorithms have been developed in the recent years to solve them. However, computationally expensive MaOPs have been scarcely investigated. Typically, surrogate-assisted methods have been used in the literature to tackle computationally expensive problems, but such studies have largely focused on problems with 1-3 objectives. In this paper, we present an approach called hybrid surrogate-assisted many-objective evolutionary algorithm to solve computationally expensive MaOPs. The key features of the approach include: 1) the use of multiple surrogates to effectively approximate a wide range of objective functions; 2) use of two sets of reference vectors for improved performance on irregular Pareto fronts (PFs); 3) effective use of archive solutions during offspring generation; and 4) a local improvement scheme for generating high quality infill solutions. Furthermore, the approach includes constraint handling which is often overlooked in contemporary algorithms. The performance of the approach is benchmarked extensively on a set of unconstrained and constrained problems with regular and irregular PFs. A statistical comparison with the existing techniques highlights the efficacy and potential of the approach.

Evolutionary algorithms have shown their promise in coping with many-objective optimization problems. However, the strategies of balancing convergence and diversity and the effectiveness of handling problems with irregular Pareto fronts (PFs) are still far from perfect. To address these issues, this paper proposes an adaptive sorting-based evolutionary algorithm based on the idea of decomposition. First, we propose an adaptive sorting-based environmental selection strategy. Solutions in each subpopulation (partitioned by reference vectors) are sorted based on their convergence. Those with better convergence are further sorted based on their diversity, then being selected according to their sorting levels. Second, we provide an adaptive promising subpopulation sorting-based environmental selection strategy for problems which may have irregular PFs. This strategy provides additional sorting-based selection effort on promising subpopulations after the general environmental selection process. Third, we extend the algorithm to handle constraints. Finally, we conduct an extensive experimental study on the proposed algorithm by comparing with start-of-the-state algorithms. Results demonstrate the superiority of the proposed algorithm.

A number of real-world problems involve extremization of multiple conflicting objectives, referred to as multiobjective optimization problems. Multiobjective evolutionary algorithms (MOEAs) have been widely adopted to obtain Pareto front (PF) approximation for such problems. An indispensable step in development and evaluation of MOEAs is benchmarking, which involves comparisons with peer algorithms using performance metrics, such as hypervolume (HV) and inverted generational distance (IGD). However, the de-facto practice is to use the final population of an algorithm for evaluating these metrics even though a better PF approximation may exist within the archive of all evaluated solutions. In a recent study, a distance-based subset selection (DSS) method was discussed for selecting prespecified number of solutions from an archive for benchmarking. This letter aims to contribute toward this direction in two ways. First is to develop a theoretical understanding of DSS and reveal some of its interesting and desirable properties. These include conditional equivalence to optimal HV/IGD subset selection, inclusion of PF extremities and invariance to convexity/concavity and orientation of the PF. Secondly, we present numerical experiments on problems with regular and irregular PFs up to ten objectives, and compare the approach with other selection techniques to demonstrate its potential benefits. The results clearly indicate the importance of considering the archive of solutions and appropriate selection mechanisms, in particular for problems with irregular PFs, to avoid misjudgments about relative performances. With increasing emphasis on tackling such problems in the field, we believe that this observation and analysis is timely and significant, not only for benchmarking, but also for subsequent improvements in decision-making and algorithm design.

Assembling a competitive team is a task encountered in many professional league sports such as cricket, soccer, rugby etc. Teams are assembled annually with players being bid for by competing franchises. While stochastic optimization approaches for team selection have been suggested in the past, the approximate nature of these techniques could be disadvantageous for team selection when the stakes are high. In this paper, we explore the use of multi-objective integer programming approach to alleviate this issue and deliver a set of optimal trade-off solutions (teams). We illustrate the performance of the approach using professional Twenty20 cricket league data from the Indian Premier League. We also demonstrate the ability to support partial team construction, i.e., selecting few members of the team with others unchanged. Lastly, we also present a way to rank the importance of the players within a team considering the key objectives.

A number of real-life optimization problems operate in hierarchical model, where a “leader” entity is trying to optimize its objective(s), while in response a “follower” entity is also optimizing its own objective(s). Such problems are referred to as bilevel optimization problems. Given their nested nature, they have additional challenges compared to the traditional single-level optimization problems. Recent studies in the domain make it evident that the number of function evaluations needed to solve such problems using evolutionary algorithms is excessive. Development of efficient techniques that could reduce this computational effort is therefore of significant interest. In our previous work, we proposed and studied the use of multiple surrogate assisted optimization (SAO) to reduce the required evaluations of the lower level problem. In this work, we extend the idea further through the use of efficient global optimization (EGO) in solving bilevel problems. We refer to the approach as BLEGO (bilevel EGO). Unlike the standard SAO techniques, EGO samples new infill locations based on expected improvement in the objective value. Two different versions of BLEGO are studied - one where EGO is used only at the lower level, and another where EGO is used at both levels. Numerical experiments are conducted on SMD benchmark problem suite and the results are compared with the previously developed surrogate assisted bilevel algorithm (SABLA) to demonstrate its efficacy.

Multiobjective optimization problems with more than three objectives are commonly referred to as many-objective optimization problems (MaOPs). Development of algorithms to solve MaOPs has garnered significant research attention in recent years. "Decomposition" is a commonly adopted approach toward this aim, wherein the problem is divided into a set of simpler subproblems guided by a set of reference vectors. The reference vectors are often predefined and distributed uniformly in the objective space. Use of such uniform distribution of reference vectors has shown commendable performance on problems with "regular" Pareto optimal front (POF), i.e., those that are nondegenerate, smooth, continuous, and easily mapped by a unit simplex of reference vectors. However, the performance deteriorates for problems with "irregular" POF (i.e., which deviate from above properties), since a number of reference vectors may not have a solution on the POF along them. While adaptive approaches have been suggested in the literature that attempt to delete/insert reference directions conforming to the geometry of the evolving front, their performance may in turn be compromised for problems with regular POFs. This paper presents a generalized version of previously proposed decomposition-based evolutionary algorithm with adaptive reference vectors, intended toward achieving competitive performance for both types of problems. The proposed approach starts off with a set of uniform reference vectors and collects information about feasibility and nondominance of solutions that associate with the reference vectors over a learning period. Subsequently, new reference directions are inserted/deleted, while the original directions may assume an active or inactive role during the course of evolution. Numerical experiments are conducted over a wide range of problems with regular and irregular POFs with up to 15 objectives to demonstrate the competence of the proposed approach with the state-of-the-art methods.

Genetic programming (GP) is one of the commonly used tools for symbolic regression. In the field of GP, the use of semantics and an external library of subexpressions for designing better search operators has recently gained significant attention. A notable example is semantic backpropagation, which has demonstrated an ability to obtain expressions with extremely small prediction errors. However, these expressions often tend to be long and difficult to interpret, which may restrict their applicability in real-life problems. In this paper, we propose a GP framework that includes two key elements, a new library construction scheme and a novel semantic operator based on mixed-integer linear programming (MILP). The proposed library construction scheme maintains diverse subexpressions and keeps the library size in check by imposing an upper limit. The proposed semantic operator constructs new expressions by effectively combining a given number of subexpressions from the library. These improvements have been integrated in a bi-objective GP framework with random desired operator (RDO), which attempts to simultaneously reduce the complexity and improve the fitness of the evolving expressions. The contributions of individual components are studied in detail using 15 benchmarks. It is observed that the use of the proposed scheme with RDO leads to shorter expressions without sacrificing accuracy of approximation. The addition of MILP further improves the results for certain types of problems.

Determining the Nash equilibria (NEs) in a competitive electricity market is a challenging economic game problem. Although finding one equilibrium has been well studied, detecting multiple ones is more practical and difficult, with a few attempts to solve such discrete game problems. However, most of the reallife game problems, such an energy market is a continuous one containing infinite sets of strategy that can be adopted by each player. Therefore, in this paper, a co-evolutionary approach is proposed for detecting multiple NEs in a single run involving continuous games among N-players. Five standard test functions and three IEEE energy market problems in three different scenarios are solved, and their results are compared with those obtained from state-of-the-art algorithms. The results clearly show the benefits of the proposed approach in terms of both the quality of solutions and efficiency.

Decomposition based approaches are increasingly being used to solve many-objective optimization problems (MaOPs). In such approaches, the MaOP is decomposed into several single-objective sub-problems and solved simultaneously guided by a set of predefined, uniformly distributed reference vectors. The reference vectors are constructed by joining a set of uniformly sampled points to the ideal point. Use of such reference vectors originating from the ideal point has so far performed reasonably well on common benchmarks such as DTLZs and WFGs, since the geometry of their Pareto fronts can be easily mapped using these reference vectors. However, the approach may not deliver a set of well distributed solutions for problems with Pareto fronts which are convex/concave or where the shape of the Pareto front is not best suited for such set of reference vectors (e.g. minus series of DTLZ and WFG test problems). While the notion of reference vectors originating from the nadir point has been suggested in the literature in the past, they have rarely been used in decomposition based algorithms. Such reference vectors are complementary in nature with the ones originating from the ideal point. Therefore, in this paper, we introduce a decomposition based approach which attempts to use both these two sets of reference vectors and chooses the most appropriate set at each generation based on the s-energy metric. The performance of the approach is presented and objectively compared with a number of recent algorithms. The results clearly highlight the benefits of such an approach especially when the nature of the Pareto front is not known a priori.

Resource constrained project scheduling problem (RCPSP) is one of the classical problems in the area of discrete optimization. In this paper we propose an algorithm for solving RCPSP which relies on an adaptive insertion mutation operator that targets different regions of the search space. Neighbourhoods are exploited via forward-backward iterative local search. Furthermore, the algorithm makes use of an archive to ensure better utilization of the schedule budget. The performance of the approach is analysed across various problem complexities associated with J30, J60 and J120 full instance sets of PSPLib with budgets of 1,000, 5,000 and 50,000 schedules. The study provides insights on the performance of the algorithm i.e. why the performance is good for particular instances and not as good for others.

Bilevel optimization, as the name reflects, deals with optimization at two interconnected hierarchical levels. The aim is to identify the optimum of an upper-level leader problem, subject to the optimality of a lower-level follower problem. Several problems from the domain of engineering, logistics, economics, and transportation have an inherent nested structure which requires them to be modeled as bilevel optimization problems. Increasing size and complexity of such problems has prompted active theoretical and practical interest in the design of efficient algorithms for bilevel optimization. Given the nested nature of bilevel problems, the computational effort (number of function evaluations) required to solve them is often quite high. In this article, we explore the use of a Memetic Algorithm (MA) to solve bilevel optimization problems. While MAs have been quite successful in solving single-level optimization problems, there have been relatively few studies exploring their potential for solving bilevel optimization problems. MAs essentially attempt to combine advantages of global and local search strategies to identify optimum solutions with low computational cost (function evaluations). The approach introduced in this article is a nested Bilevel Memetic Algorithm (BLMA). At both upper and lower levels, either a global or a local search method is used during different phases of the search. The performance of BLMA is presented on twenty-five standard test problems and two real-life applications. The results are compared with other established algorithms to demonstrate the efficacy of the proposed approach.

The field of many-objective optimization has grown out of infancy and a number of contemporary algorithms can deliver well converged and diverse sets of solutions close to the Pareto optimal front. Concurrently, the studies in cognitive science have highlighted the pitfalls of imprecise decision-making in presence of a large number of alternatives. Thus, for effective decision-making, it is important to devise methods to identify a handful (7 ± 2) of solutions from a potentially large set of tradeoff solutions. Existing measures such as reflex/bend angle, expected marginal utility (EMU), maximum convex bulge/distance from hyperplane, hypervolume contribution, and local curvature are inadequate for the purpose as: 1) they may not create complete ordering of the solutions; 2) they cannot deal with large number of objectives and/or solutions; and 3) they typically do not provide any insight on the nature of selected solutions (internal, peripheral, and extremal). In this letter, we introduce a scheme to identify solutions of interest based on recursive use of the EMU measure. The nature of the solutions (internal or peripheral) is then characterized using reference directions generated via systematic sampling and the top K solutions with the largest relative EMU measure are presented to the decision maker. The performance of the approach is illustrated using a number of benchmarks and engineering problems. In our opinion, the development of such methods is necessary to bridge the gap between theoretical development and real-world adoption of many-objective optimization algorithms.

For many real-world optimization problems, evaluating a solution involves running a computationally expensive simulation model. This makes it challenging to use evolutionary algorithms that usually have to evaluate thousands of solutions before converging. On the other hand, in many cases, even a prematurely stopped run of the simulation may serve as a cheaper, albeit less accurate (low fidelity), estimate of the true fitness value. For evolutionary optimization, this opens up the opportunity to decide about the simulation run length for each individual. In this paper, we propose a mechanism that is capable of learning the appropriate simulation run length for each solution. To test our approach, we propose two new benchmark problems, one simple artificial benchmark function and one benchmark based on a computational fluid dynamics (CFDs) simulation scenario to design a toy submarine. As we demonstrate, our proposed algorithm finds good solutions much more quickly than always using the full CFDs simulation and provides much better solution quality than a strategy of progressively increasing the fidelity level over the course of optimization.

Bilevel optimization refers to a hierarchical problem in which optimization needs to be performed at two nested levels, namely the upper level and the lower level. The aim is to identify the optimum of the upper level problem, subject to optimality of the corresponding lower level problem. Several problems from the domain of engineering, logistics, economics, and transportation have inherent nested structure which requires them to be modeled as bilevel optimization problems. Bilevel optimization usually requires inordinate amount of function evaluations since a lower level search needs to be conducted for evaluating each upper level solution. The evaluations are especially high when the problems are not suited for exact techniques and evolutionary techniques are employed instead. Reducing this computational effort has been one of the key pursuits in this domain recently. However, the use of surrogate modeling to achieve this goal has so far been scarcely studied. In this paper, we present a surrogate assisted optimization approach toward addressing this research gap. The approach uses surrogates of multiple types in order to provide flexibility of approximating different types of functions more accurately. The algorithm is further strengthened through the use of selective re-evaluation of promising solutions and periodic nested local search. The performance of the proposed algorithm is presented on twenty five standard benchmark problems. The results are compared with a number of other established evolutionary and hybrid algorithms to demonstrate the efficacy of the proposed approach in obtaining competitive results using relatively fewer function evaluations.

Design optimization of an unmanned underwater vehicle (UUV) is a complex and a computationally expensive exercise that requires the identification of optimal vehicle dimensions offering the best tradeoffs between the objectives, while satisfying the set of design constraints. Although hull form optimization of marine vessels has long been an active area of research, limited attempts in the past have focused on the design optimization of UUVs and there are even fewer reports on the use of high-fidelity analysis methods within the course of optimization. While it is understood that the high-fidelity analysis is more accurate, they also tend to be far more computationally expensive. Thus, it is important to identify when a high-fidelity analysis is required as opposed to a low-fidelity estimate. The work reported in this paper is an extension of the authors previous work of a design optimization framework, where the design problem was solved using a low-fidelity model based on empirical estimates of drag. In this paper, the framework is extended to deal with high-fidelity estimates derived through seamless integration of computer-aided design, meshing and computational fluid dynamics analysis tools i.e., computer aided 3-D interactive application, ICEM, and FLUENT. The effects of using low-fidelity and high-fidelity analyses are studied in depth using a small-scale (length nominally less than 400 mm) and light-weight (less than 450 g) toy submarine. Useful insights on possible means to identify appropriateness of fidelity models via correlation measures are proposed. The term optimality used in this paper refers to optimal hull form shapes that satisfy placement of a set of prescribed internal components.

In multiobjective engineering design, evaluation of a single design (solution) often requires running one or more computationally expensive simulation models. Surrogate assisted optimization (SAO) approaches have long been used for solving such problems, in which approximations/surrogates are used in lieu of computationally expensive simulations during the course of search. Existing SAO approaches use a variety of surrogate models and model management strategies, and the best choice is still a matter under investigation. Our current proposal is an attempt to exploit the best features of several strategies, and in particular compares two possible versions of pre-selection in multiobjective optimization. The proposed algorithm is based on the non-dominated sorting genetic algorithm (NSGA-II) but, instead of evaluating the potential offspring solutions directly, a surrogate assisted evolutionary search is conducted in the neighborhood of every offspring solution using the best local surrogate model (among Kriging, Radial basis function (RBF), Polynomial response surface method (RSM) of order 1 and 2 and Multilayer perceptrons (MLP)). Out of the combined set of candidate solutions generated using the above step, the most promising offspring solutions are pre-selected, and we examine and compare two versions of pre-selection, one ignoring the parents and one taking the parents into account. The performance of the proposed approach is studied using a number of well known numerical benchmarks and engineering design optimization problems.

Resource constrained project scheduling problem (RCPSP) is a well known problem in the area of discrete optimization. It involves scheduling a given set of activities such that they are completed within minimum possible time, while satisfying a given set of precedence and resource constraints. RCPSP has a wide applicability in a number of industries, such as engineering, management, software, etc. While the classical RCPSP has been extensively studied, literature is rather scarce when it comes to finding robust solutions to RCPSP involving uncertainties. A robust solution in this context is one whose performance is not likely to vary significantly in presence of uncertainties which are inevitable in real life scenarios, such as delays in a particular activity and/or change in the available resources. Towards addressing this gap, in this paper we formulate a variant of RCPSP with stochastic activity durations and resource availability. Further, we propose a simple population based algorithm which aims to find solutions (activity lists) with minimum average makespan in the presence of uncertainties. The output from the algorithm is compared against a chosen optimal solution for the original RCPSP in terms of robustness. We study the performance of the proposed algorithm on a number of different J30 instances taken from the widely used Project Scheduling Library (PSPLib) in order to demonstrate the utility of the approach.

High-fidelity simulations and/or physical experiments are often required to evaluate performance of products and processes. Although accurate, they are often time consuming and/or costly, which makes it prohibitive to use them within a global optimization framework. To deal with this issue, Surrogate Assisted Optimization (SAO) approaches have been commonly used in literature. Efficient Global Optimization (EGO) is one such approach which relies on Kriging model and maximizes the expected improvement to identify the best location for sampling. Most EGO approaches studied in the literature sample one point at a time. However, this approach is inefficient for the cases when it is possible to evaluate a batch of solutions at the same cost as a single solution. This might happen for example, when a number of physical samples could be tested in an experimental setup simultaneously, or a number of simulations could be run in parallel on a cluster. This study presents an approach to sample multiple locations at each iteration. To achieve this, a multi-objective (MO) formulation is proposed and solved which considers maximization of expected improvement and maximization of distance from existing truly evaluated solutions. A decomposition based approach is then used to sample the infill solutions from the resulting non-dominated front. Numerical experiments are presented on ten well known benchmarks and a comparison is done with existing methods to demonstrate the efficacy of the proposed approach. The results obtained for constrained optimization problems are particularly encouraging.

Scale-free networks are commonly used to model real-world physical and virtual systems, such as transportation networks, power grids, telecommunication networks etc. These networks are often vulnerable to malicious attacks, and making them resilient (robust) to such attacks is of significant research interest. There are various types of malicious attacks, but a particularly destructive one is the high degree adaptive attack (HDA), which sequentially targets the node(s) with highest degree in a network. Most existing approaches focus on maximizing the robustness of a base network without considering the changes required to achieve the new configuration. In this study, we first illustrate that a large improvement in the robustness of a network typically requires large changes in its configuration and thus can be costly. Subsequently, we propose a bi-objective formulation which attempts to simultaneously maximize the robustness and minimize the dissimilarity between the base and new network. A decision maker can select the most appropriate solution from the trade-off set obtained using the above formulation. The bi-objective formulation is solved using a decomposition based algorithm. Performance of the approach is illustrated on one synthetic and one real-world problem.

Bilevel optimization constitutes a specific class of problems where optimization is done at two nested levels - upper (leader) and lower (follower). The two levels are coupled by the requirement of optimality at lower level for each upper level solution. A number of real life problems in engineering, logistics, economics, transportation etc. need to be modeled as bilevel optimization problems due to involvement of a hierarchy of decision makers, and thus the problem is of significant research interest. Ensuring lower level optimality for each solution makes the problem computationally intensive in terms of number of function evaluations required. To reduce this computational effort while delivering competitive results, the authors proposed a bilevel memetic algorithm (BLMA) in their previous work, for problems with one leader and one follower. However, in a number of real-life problems, there could also exist multiple followers, which may or may not be dependent on each other. In this paper, we extend BLMA to solve problems with multiple followers. A number of benchmark multifollower problems from the literature are solved using the proposed algorithm, referred to as BLMAMF, and compared with recently reported results to demonstrate the efficacy of the approach.

Determining the optimal bidding strategies in a competitive electricity market has become an important research topic over the last few decades. In this paper, a supply function equilibrium game model is considered and formulated as a bilevel optimization problem, where the upper level is used to maximize the individual profit of each supplier and the lower one to minimize the overall operating cost. To solve this problem, a co-evolutionary approach is designed in which each supplier uses its own sub-population of a genetic algorithm to maximize its profit through a bidding strategy based on each of its generators' cost coefficients, while either a self-adaptive differential evolution or sequential quadratic programming is used to optimally allocate the generation of each supplier by minimizing the operation cost. To validate the results obtained from the proposed method, an iterative method is also used to solve the two well-known benchmarks in the literature. The results are compared with those from a state-of-art method in the literature which reveals that the co-evolutionary approach has some merits in terms of quality and reliability.

Resource constrained project scheduling problem (RCPSP) is considered to be an NP hard problem. Over the last few decades, many different approaches have been developed in order to solve RCPSPs optimally within a reasonable time limit. However, no existing approach is well-accepted in this regard. In this paper, for efficiently solving RCPSPs, a memetic algorithm is proposed. The proposed algorithm incorporates local search techniques and adaptive mutation with a carefully designed genetic algorithm. To judge the performance of the proposed algorithm, we have solved 31 benchmark problems (16 with 30 activities, and 15 problems with 60 activities), and compared the quality of solutions and computational time with other state-of-the-art algorithms. The results show that our proposed algorithm achieved good quality solutions with a significantly lower computational time.

In this paper, we introduce a novel decomposition based steady state quantum genetic algorithm for the solution of engineering optimization problems. Systematic sampling is used to generate reference directions and a small population of quantum individuals (solutions with variables represented as Q-bits) is evolved using a simple variation operator. A solution represented using Q-bits has the ability to probabilistically represent a number of solutions defined through observation. We exploit the benefits of quantum representation within a steady state evolution scheme and illustrate the behavior of the algorithm using unconstrained DTLZ2 test problem involving 2, 3, 5, and 8 objectives and a set of multi/many-objective constrained engineering design optimization problems. The underlying motivation of quantum representation stems from its ability to represent multiple states which offers the potential to evolve a small population of solutions. This aspect is magnified even further when one attempts to solve a many objective optimization problem where evolution of a large population of solutions may not be practically viable. The proposed approach is expected to gain more attention in near future as quantum computing infrastructures become more readily available.

Population based stochastic algorithms have long been used for the solution of multiobjective optimization problems. In the context of computationally expensive analysis, the existing practice utilizes some form of surrogates or approximations. In this paper, we investigate the effects of selective evaluation of promising solutions and try to derive answers to the following questions: (a) should we discard the solution based on variable values only ? (b) should we evaluate one of its objective functions and then decide to select or discard it ? or (c) should we evaluate both its objective functions before selecting or discarding it ? While evaluation of solutions is crucial for learning, it comes with a computational cost that can be significant for problems involving computationally expensive analysis. Herein, we study the effects of various selective evaluation strategies using support vector machine (SVM) classifier coupled with non-dominated sorting genetic algorithm (NSGA-II). The performance of the strategies have been evaluated using five well studied unconstrained bi-objective optimization problems (DTLZ1-DTLZ5) with limited computational budget. The results clearly indicate the benefits of using certain strategies for certain class of problems and in certain stages of the search process. Furthermore, the results also suggest that some solutions can be discarded without any evaluation, while for others after evaluation of one or both objective(s). Selective evaluation is a rarely investigated field and we hope that this study would prompt design of efficient algorithms that selectively evaluate solutions on the fly i.e. based on the trade-off between need to learn/evaluate and cost to learn.

In recent years, research in the field of bilevel optimization has gathered pace and it is increasingly being used to solve problems in engineering, logistics, economics, transportation etc. Rapid increase in the size and complexity of the problems emerging from these domains has prompted active interest in the design of efficient algorithms for bilevel optimization. While Memetic Algorithms (MAs) have been quite successful in solving single level optimization problems, there have been very few studies exploring their application in bilevel problems. MAs essentially attempt to combine advantages of global and local search strategies to locate optimum solutions with low computational cost (function evaluations). In this paper, we present a new nested approach for solving bilevel optimization problems. The presented approach uses memetic algorithm at the upper level, while a global or a local search method is used in the lower level during various phases of the search. The performance of the proposed approach is compared with two established approaches, NBLEA and BLEAQ, using SMD benchmark problem set. The numerical experiments demonstrate the benefits of the proposed approach both in terms of accuracy and computational cost, establishing its potential for solving bilevel optimization problems.

Identification of implicit and explicit relationships in a data is a generic problem commonly encountered in many fields of science and engineering. In the case of explicit relations, one is interested in identifying a compact and an accurate predictor function i.e. y = f(x), while in the implicit case, one is interested in identifying an equation of the form f(x) = 0. In both these classes of problems, one would need to search through a space of mathematical expressions, while minimizing some form of error metric. Such expressions are commonly identified using genetic programming (GP). While methods to uncover explicit equations have been studied extensively in the literature, there have been limited attempts to solve implicit cases. Since there are infinite trivial implicit forms that can be generated from a given set of data, the choice of an appropriate error metric is critical in the context of implicit equation mining. In this paper, we introduce a multiobjective genetic programming approach (MOGPA) for the solution of both classes of problems. The maximum depth of a GP-tree is used as the first objective reflecting the complexity/compactness of the expressions, while the mean error, either in the predictor variable or the implicit derivatives is used as the second objective during the course of search. The performance of the approach is illustrated using four examples. The approach delivers expressions of various complexities spanning a range of accuracy levels in a single run, unlike single objective GP formulations. It was able to identify more compact and accurate explicit forms than those from previously reported studies, and the correct, most compact expressions for implicit cases.

A Pareto Optimal Front (POF) provides the set of optimal trade-off solutions formulti-objective optimization problems. The characteristics of the POF, e.g. continuity, convexity, spread and uniformity are important in the context of decision making and performance assessment. Most of the existing metrics (hypervolume, inverted generational distance, coverage etc.) were originally designed for two or three objective optimization problems with an aim of assessing the quality of non-dominated solutions delivered by various algorithms. The metrics provide little information about the nature of the front and some of them (e.g. hypervolume) are computationally expensive for problems involving large number of objectives. For problems with more than three objectives, existing tools for visualization such as parallel plots, spider/radar plots and scatter plots also offer limited useful information in terms of the nature of the front. In this paper, we introduce an alternative scalar measure of diversity that is suitable for characterizing POF approximations of optimization problems with high number of objectives. The diversity is measured against a Reference Pareto Front, a set of points uniformly spread on the hyperplane with unit intercepts. We also illustrate that the computation of such a metric is a natural extension of decomposition based evolutionary algorithms which attempt to align solutions with the reference directions constructed using the Ideal point and uniformly distributed points on the hyperplane. In particular, the perpendicular distances between the uniformly distributed reference directions and their closest solutions in the nondominated front provides information about the diversity, while the shortest distance of every solution from the hyperplane provides information about the convexity of the POF. The proposed metrics are illustrated using a number of test problems involving up to fifteen objectives.

Decomposition-based evolutionary algorithms have been quite successful in solving optimization problems involving two and three objectives. Recently, there have been some attempts to exploit the strengths of decomposition-based approaches to deal with many objective optimization problems. Performance of such approaches are largely dependent on three key factors: 1) means of reference point generation; 2) schemes to simultaneously deal with convergence and diversity; and 3) methods to associate solutions to reference directions. In this paper, we introduce a decomposition-based evolutionary algorithm wherein uniformly distributed reference points are generated via systematic sampling, balance between convergence and diversity is maintained using two independent distance measures, and a simple preemptive distance comparison scheme is used for association. In order to deal with constraints, an adaptive epsilon formulation is used. The performance of the algorithm is evaluated using standard benchmark problems, i.e., DTLZ1-DTLZ4 for 3, 5, 8, 10, and 15 objectives, WFG1-WFG9, the car side impact problem, the water resource management problem, and the constrained ten-objective general aviation aircraft design problem. Results of problems involving redundant objectives and disconnected Pareto fronts are also included in this paper to illustrate the capability of the algorithm. The study clearly highlights that the proposed algorithm is better or at par with recent reference direction-based approaches for many objective optimization.

Robust design optimization aims to find solutions that are competent and reliable under given uncertainties. While such uncertainties can emerge from a number of sources (imprecise variable values, errors in performance estimates, varying environmental conditions, etc.), this paper focuses on problems where uncertainties emanate from design variables. In commercial designs, being reliable is often of more practical value than being globally best (but unreliable). Robust optimization poses three key challenges: 1) appropriate formulation of the problem ; 2) accurate estimation of the “robustness” measure; and 3) efficient means to identify the set of tradeoff robust solutions with an affordable computational cost. In this paper, four different problem formulations for robust optimization are presented and analyzed. The proposed formulations offer a set of tradeoff solutions with robustness from two perspectives-feasibility robustness, i.e., robustness against failure and performance robustness, i.e., robustness assuring good performance. The approach also provides means to identify critical constraints or performance functions that affect the overall robustness. The problem is posed as a many-objective optimization problem and a decomposition-based evolutionary approach is used for solving it. The performance of the proposed approach and the consequences of using different formulations are illustrated using two numerical examples and four engineering problems.

Unmanned underwater vehicles (UUVs) are becoming an attractive option for maritime search and survey operations as they are cheap and efficient compared to conventional use of divers or manned submersibles. Consequently, there has been a growing interest in UUV research among scientific and engineering communities. Although UUVs have received significant research interest in recent years, limited attention has been paid towards design and development of mini/micro UUVs (usually less than 1 foot in length). Micro unmanned underwater vehicles (μUUVs) are particularly attractive for deployment in extraordinarily confined spaces such as inspection of intricate underwater structures, ship wrecks, oil pipe lines or extreme hazardous areas. This paper considers previous work done in the field of miniature UUVs and presents an optimization framework for preliminary design of that class of UUVs. A state-of-the-art optimization algorithm namely infeasibility driven evolutionary algorithm (IDEA) is used to carry out optimization of the μUUV designs. The framework is subsequently used to identify optimal design of a torpedo-shaped μUUV with an overall length of six inches (152.4 mm). The preliminary design identified through the process of optimization is further analyzed with the help of a computer-aided design tool to come up with a detailed design. The final design has since then been built and is currently undergoing trials.

In this paper, a surrogate-assisted differential evolution (DE) algorithm is proposed to solve the computationally expensive optimization problems. In it, the Kriging model is used to approximate the objective function, while DE employs a mechanism to dynamically select the best performing combinations of parameters (amplification factor, crossover rate and population size). The performance of the algorithm is tested on the WCCI2014 competition on expensive single objective optimization problems. The experimental results demonstrate that the proposed algorithm has the ability to obtain good solutions.

Capacitated arc routing problems (CARPs) are usually modeled as static problems, where information is known in advance and assumed to remain constant during the course of optimization. However, in practice, many factors such as demand, road accessibility, vehicle availability etc. change during the course of a mission and the route of each vehicle must be reconfigured dynamically. This problem is referred to as dynamic capacitated arc routing problem (DCARP) and there have been limited attempts to solve such problems in the past. Lack of standard DCARP benchmarks is one of the key factors limiting research in this direction. This paper introduces a benchmark generator for DCARPs considering interruptions/changes that are likely to occur in realistic scenarios. These benchmarks can be used to evaluate the strengths and the weaknesses of various optimization algorithms attempting to solve realistic DCARP problems.

Capacitated arc routing problems (CARPs) are usually modeled as static problems, where all information about the problem is known in advance and assumed to remain constant during the course of optimization. However, in practice, many factors such as demand, road accessibility, vehicle availability etc. change during the course of a mission and the routes of each vehicle must be reconfigured dynamically. This problem is referred to as dynamic capacitated arc routing problem (DCARP). In this study, a memetic algorithm with a new split scheme for DCARPs is proposed. This algorithm is capable to solve DCARPs with variations in vehicle availability, road accessibility, added/canceled tasks or demands and traffic congestions. The algorithm is also capable of solving static CARPs. The performance of the algorithm is reported on a 10-node and three 100-node examples in order to demonstrate the efficacy of the algorithm in solving static and dynamic problems.

Most optimization problems in the field of engineering design involve constraints. These constraints are often due to statutory requirements (e.g. safety, physical laws, user requirements/functionality) and/or limits imposed on time and resources. Population based stochastic optimization algorithms are a preferred choice for solving design optimization problems due to their ability to deal with nonlinear black-box functions. Having a good constraint handling technique embedded within the algorithm is imperative for its good performance. With the final aim of achieving feasible optimum solutions, feasibility first techniques, i.e., those which prefer feasible solutions over infeasible, have been commonly used in the past. However, in recent studies more emphasis has been laid on intelligent use of infeasible solutions (instead of their indiscreet rejection) during the course of optimization; particularly because optimum solutions often lie on the constraint boundary. The preservation of good infeasible solutions in the population is likely to improve the convergence in constricted or disconnected feasible regions. In addition, it provides a set of marginally infeasible solutions for trade-off considerations. However, in the case of a problem consisting of a mix of hard (non-negotiable) and soft (negotiable) constraints, such trade-off solutions are practically useful if they violate the soft constraints only. In this paper, previously introduced Infeasibility Driven Evolutionary Algorithm (IDEA) is modified to deliver solutions which strictly satisfy the hard constraints and offer tradeoff solutions with respect to the soft constraints. The performance of the algorithm is demonstrated on three benchmark problems.

Engineering optimization problems often involve multiple objectives and constraints that are computed via computationally expensive numerical simulations. While the severe nonlinearity of the objective/constraint functions demand the use of population based searches (e.g. Evolutionary Algorithms), such algorithms are known to require numerous function evaluations prior to convergence and hence may not be viable in their native form. On the other hand, gradient based algorithms are fast and effective in identifying local optimum, but their performance is dependent on the starting point. In this paper, a hybrid algorithm is presented, which exploits the benefits offered by population based scheme, local search and also surrogate modeling to solve optimization problems with limited computational budget. The performance of the algorithm is reported on the benchmark problems designed for CEC 2014 Special Session and Competition on Single Objective Real-Parameter Numerical Optimization.

Over the last few decades, a number of differential evolution (DE) algorithms have been proposed with excellent performance on mathematical benchmarks. However, like any other optimization algorithm, the success of DE is highly dependent on the search operators and control parameters that are often decided a priori. The selection of the parameter values is itself a combinatorial optimization problem. Although a considerable number of investigations have been conducted with regards to parameter selection, it is known to be a tedious task. In this paper, a DE algorithm is proposed that uses a new mechanism to dynamically select the best performing combinations of parameters (amplification factor, crossover rate, and the population size) for a problem during the course of a single run. The performance of the algorithm is judged by solving three well known sets of optimization test problems (two constrained and one unconstrained). The results demonstrate that the proposed algorithm not only saves the computational time, but also shows better performance over the state-of-the-art algorithms. The proposed mechanism can easily be applied to other population-based algorithms.

This paper focuses on an optimum route search function in the in-vehicle routing guidance system. For a dynamic route guidance system (DRGS), it should provide dynamic routing advice based on real-time traffic information and traffic conditions, such as congestion and roadwork. However, considering all these situations in traditional methods makes it very difficult to identify a valid mathematical model. To realize the DRGS, this paper proposes the analytical hierarchy process (AHP) using a fuzzy inference technique based on the real-time traffic information. This AHP-FUZZY approach is a multicriterion combination system. The nature of the AHP-FUZZY approach is a pairwise comparison, which is expressed by the fuzzy inference techniques, to achieve the weights of the attributes. The hierarchy structure of the AHP-FUZZY approach can greatly simplify the definition of a decision strategy and explicitly represent the multiple criteria, and the fuzzy inference technique can handle the vagueness and uncertainty of the attributes and adaptively generate the weights for the system. Based on the AHP-FUZZY approach, a simulation system is implemented in the route guidance system, and the process is analyzed.

The performance of Differential Evolution (DE) algorithms is known to be highly dependent on its search operators and control parameters. The selection of the parameter values is a tedious task. In this paper, a DE algorithm is proposed that configures the values of two parameters (amplification factor and crossover rate) automatically during its course of evolution. For this purpose, we considered a set of values as input for each of the parameters. The algorithm has been applied to solve a set of test problems introduced in IEEE CEC'2013 competition. The results of the test problems are compared with the known best solutions and the approach can be applied to other population based algorithms.

Many objective optimization refers to the class of optimization problems involving four or more objectives. Optimal solutions of such problems lie on a hyper surface (the Pareto front) and the dimensionality of the hyper-surface is dependent on the number of objectives in conflict. Identifying well converged and well spread set of solutions spanning the hyper-surface is a non-trivial problem. In this paper we introduce a novel decomposition based steady state quantum genetic algorithm, wherein systematic sampling is used to generate the reference directions and a small population of quantum individuals (solutions with variables represented as Q-bits) is evolved using a simple variation operator. A solution represented using Q-bits has the ability to probabilistically represent a number of solutions defined through observation. We exploit the benefits of quantum representation within a steady state evolution scheme and illustrate the behavior of the algorithm using discrete formulations of the unconstrained DTLZ2 test problem involving 2, 3, 5, and 8 objectives. In order to illustrate the behavior for constrained optimization problems, we investigate the behavior of the algorithm using the water resource optimization problem involving 5 objectives. A quantum population of 5 individuals has been used to solve all the above problems. Preliminary results on the effects of the size of quantum population and the fidelity of representation are presented in this paper. Finally, a number of possible directions are suggested to further improve the performance of the algorithm.

In this paper, we discuss a practical oil production planning optimization problem. For oil wells with insufficient reservoir pressure, gas is usually injected to artificially lift oil, a practice commonly referred to as enhanced oil recovery (EOR). The total gas that can be used for oil extraction is constrained by daily availability limits. The oil extracted from each well is known to be a nonlinear function of the gas injected into the well and varies between wells. The problem is to identify the optimal amount of gas that needs to be injected into each well to maximize the amount of oil extracted subject to the constraint on the total daily gas availability. The problem has long been of practical interest to all major oil exploration companies as it has the potential to derive large financial benefit. In this paper, an infeasibility driven evolutionary algorithm is used to solve a 56 well reservoir problem which demonstrates its efficiency in solving constrained optimization problems. Furthermore, a multi-objective formulation of the problem is posed and solved using a number of algorithms, which eliminates the need for solving the (single objective) problem on a regular basis. Lastly, a modified single objective formulation of the problem is also proposed, which aims to maximize the profit instead of the quantity of oil. It is shown that even with a lesser amount of oil extracted, more economic benefits can be achieved through the modified formulation.

This paper introduces a novel identifier scheme for identification of nonlinear systems with disturbances. The identification process is carried out in two steps: an offline procedure and an online procedure. The method comprises of an automatic structure generating phase using entropybased technique. The accuracy of the model is suitably controlled using the entropy measure. The parameter learning phase uses the backpropagation technique. To improve the accuracy and also for generalization of the model to handle different data sets, Differential Evolution technique is employed whereby the parameters of the model are suitably tuned using evolutionary technique. A semi serial-parallel model is introduced to improve the online identification process in the presence of noisy data. The proposed mechanism is utilized and compared against the classical Sugeno, adaptive network-based fuzzy inference system (ANFIS) modeling and Laguerre Network-Based Fuzzy System for the identification of a nonlinear benchmark problem. In addition, the proposed technique is also used to model a rotary wing unmanned aerial vehicle (UAV) from real test input–output data. The modeling performance and generalization capability are seen to be superior with our method.

This paper proposes an efficient, adaptive constraint handling approach that can be used within the class of evolutionary multi-objective optimization (EMO) algorithms. The proposed constraint handling approach is presented within the framework of one of the most successful algorithms i.e. multi-objective evolutionary algorithm based on decomposition (MOEA/D) [1]. The constraint handling mechanism adaptively decides on the violation threshold for comparison. The violation threshold is based on the type of constraints, size of the feasible space and the search outcome. Such a process intrinsically treats constraint violation and objective function values separately and adds a selection pressure, wherein infeasible solutions with violations less than the identified threshold are considered at par with feasible solutions. As illustrated, the constraint handling scheme extends the current capability of MOEA/D to deal with constraints. The performance of the algorithm is illustrated using 10 commonly studied benchmark problems and a real-world constraint optimization problem, and compared with the results obtained using yet another commonly used form i.e. Nondominated Sorting Genetic Algorithm (NSGA-II).

Over the last few decades, a considerable number of Differential Evolution (DE) algorithms have been proposed with excellent performance on mathematical benchmarks. However, like any other optimization algorithm, the success of DE is highly dependent on its search operators and control parameters. Although a considerable number of investigations have been carried out for parameter selection, it is seen as a tedious task. In this paper, we propose a DE algorithm that uses an adaptive mechanism to select the best performing combination of parameters (amplification factor, crossover rate and the population size) during the course of a single run. The performance of the algorithm is analyzed on a set of 24 constrained optimization test problems. The results demonstrate that the proposed algorithm not only saves the computational time, but also shows better performance over the state-of-the-art algorithms.

The Evolutionary Algorithms community have had lukewarm interest in Equality constrained Multi-objective (MO) Optimization problems so far. Recently, we proposed a Most Probable Point (MPP) based repair method for equality constraint handling, where we concentrated on single-objective optimization problems. In the present work, we focus our attention to equality constrained MO optimization. We first propose a set of equality constrained MO test problems (having upto 30 variables) and then suggest a more pragmatic clustering based method for selecting the infeasible solutions to be repaired which reduces the number of function evaluations considerably. The repair procedure is integrated with the popular Evolutionary MO optimization (EMO) procedure, the NSGA-II. The results will show that the proposed procedure reaches the feasible state faster, as compared to NSGA-II for all the test problems and hence show promise as an effective method for handling equality constraints in MO optimization.

Constraint handling is an active field of research in the Genetic Algorithms community, considering that one or more constraints need to be satisfied in most real life optimization problems. Recently, we proposed a Most Probable Point based repair approach for handling equality constraints in Single-objective and Multi-objective optimization problems. In this work, we demonstrate the application of the repair approach to handle active inequality constraints. We show that the repair mechanism, which has so far been strictly applied to the domain of equality constraint handling can be used to obtain better results with faster convergence even in inequality constrained problems. We take up a number of standard Single-objective test problems having one or more active inequality constraints for our study. The applicability of the proposed procedure is demonstrated on a well studied Engineering design optimization problem. The present study contributes to the scarce body of literature available on repair mechanisms in inequality constraint handling and hence should motivate further research in this direction.

Real world optimization problems are challenging as they often involve a large number of variables and highly nonlinear constraints and objective functions. While a number of efficient optimization algorithms and numerous mathematical benchmark test functions have been introduced in recent years, the performance of such algorithms have rarely been studied across a range of real world optimization problems. In this paper, we introduce an improved adaptive differential evolution (DE) algorithm and report its performance on the newly proposed real world optimization problems. The proposed differential evolution algorithm incorporates adaptive parameter control strategies; a center based differential exponential crossover and hybridization with local search to improve its efficiency. While comprehensive results of other algorithms on the test problems are unavailable at this stage, our preliminary comparison with published results indicates promising performance of the proposed DE across the range of problems.

In this paper, an optimization framework for the design of hard chine planing craft is presented. The proposed framework consists of a surface information retrieval module, a geometry manipulation module and an optimization module backed by standard naval architectural performance estimation tools. Total resistance comprising calm water resistance and added resistance in waves is minimized subject to constraints on displacement and stability requirements. Infeasibility Driven Evolutionary Algorithm (IDEA) is incorporated in the opti mization module. A scenario-based hydrodynamic optimization problem using an example of United States Coast Guard (USCG) WPB-110ft vessel is presented in this work. The concepts presented in this paper is an extension of the works of where instead of only performing total resistance minimization of high speed planing craft at a single operational speed, a set of collective speed spanning over a predefined lifetime is illustrated. The proposed framework is capable of generating the optimum hull form while at the same time enabling a provision for ship designers to evaluate the candidate designs' performance over various operating scenarios.

Shape representation plays a vital role in any shape optimization exercise. The ability to identify a shape with good performance is largely dependent on the underlying shape representation scheme. In this paper, a novel shape representation scheme is presented based on B-splines, wherein the control points representing the shape are repaired and subsequently evolved within the framework of a memetic algorithm. The underlying memetic algorithm is a multi-feature hybrid that combines the strength of a real coded genetic algorithm, differential evolution and a local search. Two test problems on shape matching are presented and solved using a mere 5000 function evaluations to illustrate the efficiency of the proposed scheme.

Genetic Algorithms (GAs) have been studied for more than three decades now. Their application in optimization problems is well understood and significant amount of research has gone into the development of efficient GA operators. Besides GAs, a number of other Evolutionary Algorithms (EAs) and their performance-enhancing variations have been proposed. However, this upgraded performance is often achieved at the undesirable cost of introducing additional user-defined parameters. In an attempt to put forward a case for GA even when a plethora of other EAs are available, we present the results obtained by using a Real-Coded, Elite preserving GA on the Real World optimization problems of IEEE CEC - 2011. Based on our preliminary investigations, we would like to stress that the current work shall help bring forth the need to take a step back to re-assess the applicability of basic GAs to practical optimization before yet another Bio-inspired algorithm is introduced.

Multi-objective optimization methods focus towards finding the high-performing Pareto-optimal solutions, without considering their sensitivity to minor deviations from their original values. It is a fair assumption that practical realization of optimal solutions is often accompanied by minor differences from the exact numerical results produced by an optimizer. Taking this factor into account, Robust Optimization methods seek to find high-performing solutions which are also less sensitive to such deviations. In this work, we have proposed strategies to minimize the number of function evaluations (which can be an expensive enterprise) to enhance one of the earliest proposed methods for robust Multi-objective Optimization. Our focus is on constrained Multi-objective optimization problems and hence we make use of the Infeasibility Driven Evolutionary Algorithm (IDEA), as the Evolutionary Multi-objective Optimizer. We take up three constrained Multi-objective engineering design optimization problems from the literature as the test-bed for our experiments and present results on the same.

Evolutionary Algorithms (EAs), in their traditional form or in combination as in Memetic Algorithms (MAs), have been quite successful in solving a variety of optimization problems in the past. More recently, several excellent Differential Evolution (DE) based algorithms have been proposed which have had outstanding success in IEEE Congress on Evolutionary Computation (CEC) competition problems. Inspired by previous studies, we propose an algorithm combining the strengths of EA, DE and MA in this paper. The algorithm utilizes a population of random solutions to start with and generates a child population either through EA or DE based evolution with equal probability. Local search is then performed from one of the solutions in the population for further improvement objective value. To avoid stagnation, re-initialization of the population is performed whenever the local search is unable to improve the values consecutively for a prescribed number of generations. The performance of the proposed algorithm is presented in this paper for the newly introduced real world optimization problems for CEC 2011 competition.

Many-objective optimization refers to the optimization problems containing large number of objectives, typically more than four. Non-dominance is an inadequate strategy for convergence to the Pareto front for such problems, as almost all solutions in the population become non-dominated, resulting in loss of convergence pressure. However, for some problems, it may be possible to generate the Pareto front using only a few of the objectives, rendering the rest of the objectives redundant. Such problems may be reducible to a manageable number of relevant objectives, which can be optimized using conventional multiobjective evolutionary algorithms (MOEAs). For dimensionality reduction, most proposals in the paper rely on analysis of a representative set of solutions obtained by running a conventional MOEA for a large number of generations, which is computationally overbearing. A novel algorithm, Pareto corner search evolutionary algorithm (PCSEA), is introduced in this paper, which searches for the corners of the Pareto front instead of searching for the complete Pareto front. The solutions obtained using PCSEA are then used for dimensionality reduction to identify the relevant objectives. The potential of the proposed approach is demonstrated by studying its performance on a set of benchmark test problems and two engineering examples. While the preliminary results obtained using PCSEA are promising, there are a number of areas that need further investigation. This paper provides a number of useful insights into dimensionality reduction and, in particular, highlights some of the roadblocks that need to be cleared for future development of algorithms attempting to use few selected solutions for identifying relevant objectives.

Real life optimization problems often involve a tradeoff between multiple objectives. They also comprise constraints, which may be due to various factors such as the strength of materials, stability and safety of design, limits on operational time, financial viability and many others. In recent years, Evolutionary Algorithms (EAs) have been a popular choice for solving such multi-objective, constrained problems for various reasons. Point to point search methods, such as Simulated Annealing (SA) have been largely unexplored for such problems. However, some recent studies have suggested that with certain enhancements, SA can also be an effective tool to deal with such problems. The ability of SA to accept uphill moves (unlike EAs) during the search makes it a less prone to getting trapped in a local minimum, and is therefore attractive for solving optimization problems. For many engineering design problems, evaluating a candidate solution can be computationally intensive. This can often put a restriction on the number of function evaluations to find a near optimum solution. To mitigate this problem, often suitable approximations (surrogates) are used in optimization instead of real function evaluations. Surrogate assisted approaches have so far been used in the paradigm of Evolutionary Algorithms (EA). In this paper, we extend the use of surrogates to another well known heuristic, Simulated Annealing (SA), for constrained, multi-objective problems. A comparison with currently prevalent EAs on a difficult set of constrained problems has been included in order to highlight the efficacy of the proposed approach.

Real life optimization problems often involve one or more constraints, and there is a significant interest among the research community to develop efficient algorithms to solve such constrained optimization problems. This paper presents a memetic algorithm combining the strengths of an evolutionary algorithm and a local search strategy. Since solutions of constrained optimization problems are expected to lie on constraint boundaries for most problems, the algorithm explicitly preserves marginally infeasible solutions to intensify search around the constraint boundaries. Furthermore, local search is done from solutions within the population to yield good quality solutions in early generations. The concepts of injecting high quality solutions in earlier generations and preservation of marginally infeasible solutions are both known to improve the efficiency of evolutionary algorithms for constrained optimization problems. The performance of the algorithm is presented on the newly proposed set of test functions (C01-C18)for 10 and 30 dimensions.

Dynamic multi-objective optimization (DMO) is a challenging class of problems where the objective and/or the constraint function(s) change over time. DMO has received little attention in the past and none of the existing multi-objective optimization algorithms have performed too well on the set DMO test problems. In this paper, we introduce a memetic algorithm (MA) embedded with a sequential quadratic programming (SQP) solver for faster convergence and an orthogonal epsilon-constrained formulation is used to deal with two objectives. The performance of the memetic algorithm is compared with an evolutionary algorithm (EA) embedded with a Sub-EA with and without restart mechanisms on two benchmark functions FDA1 and modified FDA2. The memetic algorithm consistently outperforms the evolutionary algorithm for both FDA1 and modified FDA2 problems.

A cooperative coevolutionary algorithm (CCEA) is an extension to an evolutionary algorithm (EA); it employs a divide and conquer strategy to solve an optimization problem. In its basic form, a CCEA splits the variables of an optimization problem into multiple smaller subsets and evolves them independently in different subpopulations. The dynamics of a CCEA is far more complex than an EA and its performance can vary from good to bad depending on the separability of the optimization problem. This paper provides some insights into why CCEA in its basic form is not suitable for nonseparable problems and introduces a cooperative coevolutionary algorithm with correlation based adaptive variable partitioning (CCEA-AVP) to deal with such problems. The performance of CCEA-AVP is compared with CCEA and EA to highlight its benefits. CCEA-AVP offers the possibility to deal with problems where separability among variables might vary in different regions of the search space.

Many objective optimization is a natural extension to multi-objective optimization where the number of objectives are significantly more than five. The performance of current state of the art algorithms (e.g. NSGA-II, SPEA2) is known to deteriorate significantly with increasing number of objectives due to the lack of adequate convergence pressure. It is of no surprise that the performance of NSGA-II on some constrained many-objective optimization problems (Deb and Saxena, 2006) (e.g., DTLZ5-(5,M), M = 10, 20) in an earlier study (Saxena, 2008) was far from satisfactory. Till date, research in many-objective optimization has focussed on two major areas (a) dimensionality reduction in the objective space and (b) preference ordering based approaches. This paper introduces a novel evolutionary algorithm powered by epsilon dominance (implemented within the framework of NSGA-II) and controlled infeasibility for improved convergence while the critical set of objectives is identified through a nonlinear dimensionality reduction scheme. Since approaching the Pareto-optimal front from within the feasible search space will need to overcome the problems associated with low selection pressure, the mechanism to approach the front from within the infeasible search space is promising as illustrated in this paper. The performance of the proposed algorithm is compared with NSGA-II (original, with crowding distance measure) and NSGA-II (epsilon dominance) on the above set of constrained multiobjective problems to highlight the benefits.

A number of population based optimization algorithms have been proposed in recent years to solve unconstrained and constrained single and multi-objective optimization problems. Most of such algorithms inherently prefer a feasible solution over an infeasible one during the course of search, which translates to approaching the constraint boundary from the feasible side of the search space. Previous studies [1], [2] have already demonstrated the benefits of explicitly maintaining a fraction of infeasible solutions in Infeasiblity Driven Evolutionary Algorithm (IDEA) for single and multiobjective constrained optimization problems. In this paper, the benefits of IDEA as a sub-evolve mechanism are highlighted for dynamic, constrained single objective optimization problems. IDEA is particularly attractive for such problems as it offers a faster rate of convergence over a conventional EA, which is of significant interest in dynamic optimization problems. The algorithm is tested on two new dynamic constrained test problems. For both the problems, the performance of IDEA is found to be significantly better than conventional EA.

Many objective optimization refers to optimization problems for which the number of objectives is significantly greater than conventionally studied 2 or 3. For such problems, large number of solutions become non-dominated, which reduces the convergence pressure of the Evolutionary Algorithms~(EAs) towards the Pareto Optimal Front. Recently, alternate secondary ranking schemes for have been suggested for NSGA-II in lieu of crowding distance to expedite its convergence for many objective problems. In this paper, we improvise upon an existing scheme~(epsilon dominance). The proposed approach is found to perform better than the other substitute distance assignment methods for the problems studied in this paper. A new diversity metric has also been proposed, which can be used in order to compare the performance of the various EAs.

The most common approach to handling constraints in a constrained optimization problem has been the use of penalty functions. In recent years non-dominance based ranking methods have been applied for an efficient handling of constraints. These techniques favor the feasible solutions over the infeasible solutions, thus guiding the search through the feasible space. Usually the optimal solutions of the constrained optimization problems are spread along the constraint boundary. In this paper we propose a constraint handling method that maintains infeasible solutions in the population to aid the search of the optimal solutions through the infeasible space. The constraint handling method is implemented in constraint handling evolutionary algorithm (CHEA), which is the modified non-dominated sorting genetic algorithm II (NSGA-II) [1]. The original constrained minimization problem with k objectives is reformulated as an unconstrained minimization problem with k + 1 objectives, where an additional objective function is the number of constraint violations. In CHEA, the infeasible solutions are ranked higher than the feasible solutions, thereby focusing the search for the optimal solutions near the constraint boundaries through infeasible region. CHEA simultaneously obtains the solutions to the constrained as well as the unconstrained optimization problem. The performance of CHEA is compared with NSGA-II on the set of CTP test problems. For a fixed number of function evaluations, CHEA converges to the Pareto optimal solutions much faster than NSGA-II. It is observed that retaining even a small number of infeasible solutions in the population, CHEA is able to prevent the search from prematurely converging to a sub-optimal Pareto front.

In this paper, we introduce a simulated annealing algorithm for constrained Multi-Objective Optimization (MOO). When searching in the feasible region, the algorithm behaves like recently proposed Archived Multi-Objective Simulated Annealing (AMOSA) algorithm [1], whereas when operating in the infeasible region, it tries to minimize constraint violation by moving along Approximate Descent Direction (ADD) [2]. An Archive of non-dominated solutions found during the search is maintained. The acceptance probability of a new point is determined by its feasibility status, and its domination status as compared to the current point and the points in the Archive. We report the performance of the proposed algorithm on a set of seven constrained bi-objective test problems (CTP2 to CTP8), which have been known to pose difficulties to existing multi-objective algorithms. A comparative study of current algorithm with the widely used multi-objective evolutionary algorithm NSGA-II has been included.

In this paper, we introduce a simulated annealing algorithm for single objective, trans-dimensional optimization problems. Trans-dimensional optimization refers to a class of problems where candidate solutions can have different number of variables. For such problems, the existing optimization methods need to be run for various models (i.e. problems with fixed number of variables) extensively, which is inefficient. The proposed optimization algorithm explores the model space and the corresponding variable space probabilistically, allocating more computational resources (function evaluations) to the promising models. The performance of the proposed algorithm is reported for a clustering problem and a warehouse optimization problem. The results of the proposed algorithm are compared with a conventional optimization algorithm with fixed number of variables (NSGA-II [3]) to highlight the benefits of the approach.

Dynamic multi-objective optimization (DMO) is one of the most challenging class of optimization problems where the objective functions change over time and the optimization algorithm is required to identify the corresponding Pareto optimal solutions with minimal time lag. DMO has received very little attention in the past and none of the existing multi-objective algorithms perform satisfactorily on test problems and a handful of such applications have been reported. In this paper, we introduce a memetic algorithm (MA) and illustrate its performance for online neural network (NN) identification of the multi-input multi-output unmanned aerial vehicle (UAV) system. As a typical case, the longitudinal model of the UAV is considered and the performance of a NN trained with the memetic algorithm is compared to another trained with Levenberg-Marquardt training algorithm using mini-batches. The memetic algorithm employs an orthogonal epsilon-constrained formulation to deal with multiple objectives and a sequential quadratic programming (SQP) solver is embedded as its local search mechanism to improve the rate of convergence. The performance of the memetic algorithm is presented for two benchmarks Fisherpsilas Discriminant Analysis (FDA), FDA1 and modified FDA2 before highlighting its benefits for online NN model identification for UAVs. Observations from our recent work indicated that Mean Square Error (MSE) alone may not always be a good measure for training the networks. Hence the MSE and maximum absolute value of the instantaneous error is considered as objectives to be minimized which requires a Dynamic MO algorithm. The proposed memetic algorithm is aimed to solve such identification problems and the same can be extended to control problems.

Presented in this paper is a hybrid algorithm simplex search enabled evolutionary algorithm (SSEA) which is fundamentally an evolutionary algorithm (EA) embedded with a local simplex search for unconstrained optimization problems. Evolutionary algorithms have been quite successful in solving a wide class of intractable problems and the non-dominated sorting genetic algorithm (NSGA-II) is a popular choice. However, like any other evolutionary algorithms, the rate of convergence of NSGA-II slows down with generations and often there is no improvement in the best candidate solution over a number of generations. The simplex search component comes into effect once the basic evolutionary algorithm encounters a slow rate of convergence. To allow exploitation around multiple promising regions, the simplex search is invoked from multiple promising regions of the variable space identified using hierarchical agglomerative clustering. In this paper, results are presented for a series of unconstrained optimization test problems that cover problems with a single minimum, a few minima and a large number of minima. Provided is a comparison of results with NSGA-II, fast evolutionary strategy (FES), fast evolutionary programming (FEP) and improved fast evolutionary programming (IFEP) where it's clear that SSEA outperforms all other algorithms for unimodal problems. On the suite of problems with large number of minima, SSEA performs better on some of them. For problems with fewer minima, SSEA performs better than FES, FEP and IFEP while demonstrating comparable performance to NSGA-II.

Presented in this paper is a novel scheme of representation for truss geometry. Trusses are represented as a set of elements having a collection of properties (e.g. cross-sectional area, type of material). These sets can be of varying cardinality representing truss structures with different numbers of elements and hence distinctly different topologies. A recombination operator to handle a set representation that can generate offspring topologies that can be different from the parents is also proposed. Depending on the physical problem being solved, one can introduce specific operators which will aid the optimization process. One such mutation operator is used for the truss design to reduce the number of elements, hence finding the smallest feasible topology for the truss structure. Another mutation operator perturbs the properties of the elements using the Gaussian mutation.

Path planning is the process of generating a path between an initial location and a target location that has optimal performance against specific criteria. This paper addresses the problem of offline path planning as applied to autonomous miniature fixed wing unmanned aerial vehicles (mini-UAVs). The path representation takes into account aircraft dynamics by incorporating the turn rates and velocities of the UAV and follows a waypoint guidance method that is adopted in commercial aviation industry. The aircraft dynamics model allows the computation of fuel use, throttle, and velocity at different time instants throughout the path. A rigorous model validation is carried out prior to using the model for optimal path identification. An evolutionary algorithm is used to optimize the path distance and threat exposure encountered by the UAV for a mission. The optimization algorithm is a stochastic, zero order, elitist method similar in many respects to nondominated sorting genetic algorithm (NSGA-II) but includes explicit diversity maintaining mechanism in both the objective and variable space. A number of case studies are included to highlight the benefits offered by our approach.

Machine layout and material flow between machines are crucial considerations for improving productivity in any manufacturing environment. The machine layout and the operations assignment problems are both known to be NP hard problems. In this paper, we introduce a new combined machine layout and operations assignment problem. We propose an evolutionary algorithm to solve the combined machine layout and operations assignment problem. The effectiveness of our approach is demonstrated through numerical examples.

In this paper, we discuss a practical oil production problem from a petroleum field. A field typically consists of a number of oil wells and to extract oil from these wells, gas is usually injected which is referred as gas-lift. The total gas used for the oil extraction is constrained by daily availability limits. The oil extracted from each well is known to be a nonlinear function of the gas injected into a well and varies between wells. The problem is to identify the optimal amount of gas that needs to be injected into each well to maximize the amount of oil extracted subject to the constraint mentioned earlier on a daily basis. The problem has long been of practical interest to all major oil exploration companies as it has a potential of deriving large financial benefits. Considering the complexity of the problem, we have used an evolutionary algorithm to solve the production planning problem. The multiobjective formulation is attractive as it eliminates the need to solve such problems on a daily basis while maintaining the quality of solutions. Our results show significant improvement over the existing practices.

We report the results of our study on the behaviour and implementation of the parent centric operator (PCX) within the generalized generation gap (G3) model using five test functions of 10, 20 and 50 dimensions. Our study indicates that G3-PCX performs fairly well on most functions, but its performance is not good for highly nonlinear, multidimensional problems (Rastrigin, Ackley, Griewangk). We observed the same behaviour of G3-PCX while designing a 22 element Yagi-Uda Antenna for gain maximization (known to be a highly nonlinear problem). We derived a simple variant G3-PCX-II using a Roulette wheel based parent selection scheme which performs better than G3-PCX on the highly nonlinear multidimensional problems.

We report the behavior of kriging and cokriging based surrogate models within the optimization framework. The framework is built upon a stochastic, zero order, population-based optimization algorithm embedded with controlled elitism to ensure convergence in the actual function space. The model accuracy is maintained via periodic retraining and the number of data points required to create the surrogate model is adaptively identified using Calinski Harabasz (CH) index. Results of kriging and cokriging are compared with radial basis function models on a set of numerical and engineering design optimization problems.

Design of Yagi-Uda antennas is a challenging problem since antenna characteristics such as gain, input impedance, maximum sidelobe level etc., are known to be extremely sensitive to the design variables viz., element lengths and their spacings. Although, population-based, stochastic, zero-order methods like genetic algorithm (GA) and evolutionary algorithm (EA) are attractive choices for such classes of problems, their successful application requires a number of additional inputs (e.g. scaling and aggregating factors to deal with constraints and objectives) that is not easy for a designer to provide. We introduce a population-based, stochastic, zero-order optimization algorithm and use it to solve single and multiobjective Yagi Uda design optimization problems. The algorithm is attractive as it is computationally efficient and does not require additional user inputs to model constraints or objectives. One single objective and two multiobjective Yagi Uda design examples are presented. The first example highlights the limitations of using an aggregate objective function in design optimization, while the second and the third examples illustrate the performance of our optimization algorithm for multiobjective problems.

Population-based, stochastic, zero-order optimization methods (e.g. genetic and evolutionary algorithms) are a popular choice in solving intractable, real-life optimization problems. These methods are particularly attractive as they are easy to use and do not require assumptions about functional and slope continuities unlike some of its gradient-based counterparts. Despite their advantages, these methods require the evaluation of numerous candidate solutions, which is often computationally expensive and practically prohibitive. We introduce a framework for optimization using approximate functions. The optimization algorithm is a population-based, stochastic, zero-order, elite-preserving algorithm that makes use of approximate function evaluations in lieu of actual function evaluations. The approximate function is constructed using a radial basis function (RBF) network and the network is periodically retrained after a few generations unlike other models which create and use the same approximate model repeatedly without retraining. A scheme for controlled elitism is incorporated within the optimization framework to ensure convergence in the actual function space. The computational accuracy and efficiency of the proposed optimization framework is assessed using a set of five mathematical test functions. The results clearly indicate that the optimization framework using approximations is able to arrive at reasonably accurate results using only a fraction of actual functions evaluations.

The ability to mutually interact is a fundamental social behavior in all human and insect societies. Social interactions enable individuals to adapt and improve faster than biological evolution based on genetic inheritance alone. This is the driving concept behind the optimization algorithm introduced in this paper that makes use of the intra and intersociety interactions within a formal society and the civilization model to solve single objective constrained optimization problems. A society corresponds to a cluster of points in the parametric space while a civilization is a set of all such societies. Every society has its set of better performing individuals (leaders) that help others to improve through information exchange. This results in the migration of a point toward a better performing point, analogous to an intensified local search. Leaders improve only through an intersociety information exchange that results in the migration of a leader from a society to another. This helps the better performing societies to expand and flourish.

A major fraction of evolutionary optimization methods aims to find solutions that maximize performance. However, a solution that solely maximizes performance is of no practical use as it may be too sensitive to parametric variations (nonuniform material properties, inexact physical dimensions, uncertainties in loading and operating conditions, etc.). Furthermore, for design problems with constraints, a robust solution needs to be feasible and remain feasible under parametric variations. In this paper, a new evolutionary algorithm is proposed that is capable of handling constrained robust optimal design problems. A multiobjective formulation is introduced that considers an individuals' performance, the mean performance of its neighbors and the standard deviation of its neighbors' performance as three objectives for optimization. In order to handle feasibility, an innovative constraint-handling scheme based on the Pareto concept is introduced that considers an individual's self-feasibility and its neighborhood feasibility. Robust optimal solutions to two engineering design examples are reported in this paper. Results of simulations are also presented to illustrate the differences between an optimal solution and a robust optimal solution.

We present an effective multilevel information sharing strategy within a swarm to handle single objective, constrained and unconstrained optimization problems. A swarm is considered as a collection of individuals having a common goal to reach the best value (minimum or maximum) of a function. The success of a swarm is attributed to the identification of a set of competent leaders and a meaningful information sharing scheme between the leaders and the rest of the individuals that enables the swarm to collectively attain the common goal. The proposed algorithm mimics the above behavioral processes of a real swarm and maintains unique individuals at all time instants. The uniqueness among the individuals result in a set of near optimal solutions at the final phase that is useful for sensitivity analysis. The benefits of the effective information sharing strategy is illustrated by solving two unconstrained problems with multiple equal and unequal optima and a constrained optimization problem.