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.

Practical optimization problems may contain differ-ent kinds of difficulties that are often not tractable if one relies on a particular optimization method. Different optimization approaches offer different strengths that are good at tackling one or more difficulty in an optimization problem. For instance, evolutionary algorithms have a niche in handling complexities like discontinuity, non-differentiability, discreteness and non-convexity. However, evolutionary algorithms may get computationally expensive for mathematically well behaved problems with large number of variables for which classical mathematical programming approaches are better suited. In this paper, we demonstrate a decomposition strategy that allows us to synergistically apply two complementary approaches at the same time on a complex optimization problem. Evolutionary algorithms are useful in this context as their flexibility makes pairing with other solution approaches easy. The decomposition idea is a special case of bilevel optimization that separates the difficulties into two levels and assigns different approaches at each level that is better equipped at handling them. We demonstrate the benefits of the proposed decomposition idea on a wide range of test problems.

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.

No abstract available.

We present a modified planning-assisted swarm shepherding method to effectively control multi-robot (sheepdogs) when herding a swarm of reactive agents (sheep) towards a goal and in environments with obstacles. Given a highly-dispersed sheep swarm, a mission planner based on Ant Colony Optimisation and A * is designed and developed to support the shepherding task. To apply the swarm shepherding method to real robots, a multi-layer environmental modelling method is proposed to construct customised environment maps for sheep and sheepdogs according to their physical characteristics. Then, a lookahead - based sub-goal selection method is presented for herding the sheep swarm to follow the A * optimised reference path. Furthermore, a circle-based method for selecting feasible driving/collecting points while avoiding obstacles is designed. Experiments are conducted in numerical simulation environments to compare the proposed method with the state-of-the-art planning-assisted shepherding method, followed by testing in the robot simulation platform CoppeliaSim to demonstrate the effectiveness of the proposed method.

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.

Shepherding is a nature-inspired swarm guidance approach, where one or more sheepdogs act as actuators to guide a swarm towards a goal area. In the real-world, swarm guidance occurs in unknown environments. Context unfolds as the controller agent, the sheepdog, continues to discover new states, causing the state space to unfold during a mission due to the partial observability of the state space by each sheepdog. These individualised experiences could get shared among the shepherds to improve situation awareness. Our prior work introduced an approach to share interpretable knowledge between two agents. In this paper, we extend the two-agent interpretable knowledge fusion algorithm to multi-agent settings; allowing multiple sheepdogs to share their knowledge in an interpretable manner. When an agent receives knowledge from another agent, it decides on whether to integrate this new knowledge with what it already knows, leading to an increase in the size and space complexity of an agent's knowledge base. We propose a modular neural network society of mind architecture to store, update and manipulate the knowledge base of an agent. The architecture stores sub-networks and associate them with situations. When an agent is faced with a state, a gate controller decides based on the situation facing the agent which sub-networks are best suited to make decisions. The contribution is validated on a general classification task utilising the full-state-space for a swarm guidance shepherding problem. When compared to baseline methods, the proposed knowledge transfer algorithm improves generalisation, reduces catastrophic forgetting, and produces smaller models with faster adaptation.

This paper investigates a swarm shepherding problem which aims to herd multiple sub-swarm of robot agents (sheep) in a large-scale cluttered environment to a specific goal area using multiple distance-constrained robots (sheepdogs) located at different depots. We propose to formulate this challenging problem as a Multi-depot, Distance-constrained Close-Open Mixed Vehicle Routing Problem (MDCOMVRP). We also design a Two-phase Ant Colony Optimisation to address it by decomposing MDCOMVRP into a Multi-depot Open Vehicle Routing Problem (MOVRP) and a split problem. In the first phase, the Max-Min Ant System algorithm is employed to find open routes for all robots by transforming the MOVRP into a standard Travelling Salesman Problem using the proposed transformation method. In the second phase, a Modified Split algorithm is presented to construct a set of close or open distance-constrained routes, which are further optimised by the 2-opt local search method to generate the optimised sequence for each sheepdog robot to collect/drive sheep sub-swarms. Experiments are conducted to demonstrate that the proposed algorithm can solve MDCOMVRP successfully and assist the robots to complete the swarm shepherding mission efficiently.

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.

Controlling a large swarm of agents is a challenging task. Shepherding refers to an active field of research that seeks to address this challenge by using a control agent (sheepdog), which guides a swarm (sheep) towards a goal. Traditional shepherding involves switching between two main behaviours: driving the swarm towards the goal, and collecting stray sheep back to the flock. Evidently, the movement of the agents are dependent on their sensed information. Therefore, effectively controlling a swarm is even more challenging when sensor information or communication channels are unreliable. In this paper, we propose a shepherding methodology to achieve efficient swarm control in the presence of noise in the sensed information. The proposed approach consists of a new resting behaviour and a neural network-based reinforcement learning model. The neural network is used to learn shepherding policies using the new resting behaviour, where the objective is to optimise the frequency of sheep-to-dog interactions with varying levels of noise. The proposed approach is validated through simulations. Numerical experiments show that the proposed approach results in a more effective and stable performance compared to some conventional shepherding models from the literature.

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.

Hypervolume (HV) is one of the most commonly used metrics for evaluating the Pareto front (PF) approximations generated by multiobjective evolutionary algorithms. Even so, HV is a resultant of a complex interplay between the PF shape, number of objectives, and user-specified reference points which, if not well understood, may lead to misinformed inferences about benchmarking performance. In order to understand this behavior, some previous studies have investigated such interactions empirically. In this letter, a new and unconventional approach is taken for gaining further insights about HV behavior. The key idea is to develop theoretical formulas for certain linear (equilateral simplex) and quadratic (orthant) PFs in two specific orientations: 1) regular and 2) inverted. These PFs represent a large number of problems in the existing DTLZ and WFG suites commonly used for benchmarking. The numerical experiments are presented to demonstrate the utility of the proposed work in benchmarking, and in understanding the contributions of the different regions of the PFs, such as corners, edges, as well explaining the contrast between the HV behaviors for regular versus inverted PFs. This letter provides a foundation and computationally fast means to undertake parametric studies to understand various aspects of HV.

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.

In the shepherding problem, an external agent (the shepherd) attempts to influence the behavior of a swarm of agents (the sheep) by steering them towards a goal that is known to the shepherd but not the sheep. The problem offers a level of abstraction for Human-Swarm Interaction, where the human is able to shepherd the swarm towards a goal. Similarly, a smart robot could act as a shepherd to replace biological shepherds with ground or air vehicles. In both cases, it is important to preserve the energy of the shepherd by modulating the shepherd's influence vector on the sheep. Therefore, in this paper, we design a force modulation function for the shepherd agent to optimize the energy used by the agent and systematically study the effect of modulating the force of the influence vector on task success and energy used. The problem is further investigated using a bi-objective optimization formulation, where the energy used by the shepherd as well as the time of completion of the task are minimized, subject to a threshold of success rate. The findings demonstrate the coupling between contextual information used by the shepherd to modulate its influence vector and the effectiveness and efficiency of shepherd to complete the task.

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.

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.

Real-world optimization problems often involve constraints that relate to viability of implementing a solution. To solve such problems efficiently, a good constraint handling method is indispensable for an optimization algorithm. Population-based optimization algorithms allow a flexible way to handle constraints by making a careful comparison between feasible and infeasible solutions present in the population. A previous approach, which emphasized feasible solutions infinitely more than the infeasible solutions, has been popularly applied for more than one-and-half decade, mostly with real-parameter genetic algorithms (RGAs). Despite its popular use, the idea was criticized for its extreme selection pressure against infeasible solutions. Since optimal solutions often lie on the constraint boundaries, survival of certain infeasible solutions close to critical constraint boundaries should help RGA's recombination and mutation operators to produce near-optimal solutions. In this paper, we extend the earlier parameter-less constraint handling approach so as to strike a balance between survival of feasible and infeasible solutions in a GA population. The balance is controlled through an additional parameter that could be pre-specified or adaptively updated as the algorithm progresses. A parametric study is conducted to determine an appropriate value which works the best on most problems of this study. A significant improvement in performance is observed for the commonly-used g-series test problem suite and a real-world application problem (welded beam design). The approach is generic and can be easily extended to other real-parameter evolutionary algorithms, multi-objective and other advanced optimization tasks.

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.

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.

The multiobjective evolutionary algorithm based on decomposition (MOEA/D) converts a multiobjective optimization problem into a set of single-objective subproblems, and tackles them simultaneously. In MOEA/D, the offspring generation is a crucial part to increase the convergence of the algorithm and maintain the diversity of the solution set. Currently, the majority of reproduction operators consider the quality of neighborhood exploration, i.e., the capability to distribute along the population structure, while few operators have good capability for subproblem exploitation, i.e., the ability to push solutions forward along the subproblems. To address this issue in this paper, we introduce one of the derivative-free optimization methods, Nelder-Mead simplex (NMS) method, to MOEA/D to accelerate the algorithm convergence. The NMS operator is combined with a differential evolution (DE) operator in the offspring generation. The comparison study demonstrates that calling the NMS operator occasionally can help to accelerate the convergence.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.