In evolutionary multi-objective optimization (EMO), a set of well-distributed non-dominated solutions is typically obtained by an EMO algorithm. However, the choice of a final solution from the obtained solution set has not been discussed in many studies. Moreover, for standard EMO algorithms without preference information, it has rarely been discussed whether the obtained solution set includes a sufficient number of well-balanced trade-off solutions over all objectives for selection by a decision maker. Since the final goal of multi-objective optimization is to find the most preferred solution for the decision maker, it is necessary for EMO algorithms to find those promising solutions. From this viewpoint, one issue in many EMO algorithms is that the obtained solution sets usually include many solutions on the boundary of the Pareto front, especially in the case of many-objective optimization. Since boundary solutions are not balanced over all objectives, they are less likely to be selected as a final solution by the decision maker. To address this issue, we propose a modified MOEA/D algorithm with an interior weight vector generation strategy after discussing the negative effects of boundary weight vectors. Experimental results demonstrate the effectiveness of the proposed approach in finding balanced and promising solutions.

Subset selection is a key task in evolutionary multi-objective optimization. Addressing this combinatorial challenge is key to generating finite μ-point Pareto front approximations (PFAs) that accurately represent the Pareto front, regardless of its geometric shape and dimension. For instance, subset selection is used both in bounded-size archiving and in the environmental selection of an EMO algorithm. A significant challenge remains in designing algorithms for medium-and large-scale subset selection instances such as those that involve unbounded external archives (UEA). In this article, we propose two efficient algorithms based on Riesz s-energy (RSE). These algorithms employ two main strategies: greedy inclusion and iterative replacement, exhibiting linear time and space complexity for medium-and large-scale subset selection instances. Our experimental results show that our RSE-based algorithms produce target subsets with high diversity at a low processing time, outperforming six state-of-the-art subset selection algorithms. Additionally, we validated their effectiveness in extracting a representative PFA from a UEA connected to a multi-objective evolutionary algorithm, producing well-diversified subsets regardless of the Pareto front geometry and its dimension.

Abstract Indicator-based (multi-objective) diversity optimization aims at finding a set of near (Pareto)optimal solutions that maximizes a diversity indicator, where diversity is typically interpreted as the number of essentially different solutions. Whereas, in the first diversity-oriented evolutionary multi-objective optimization algorithm, the NOAH algorithm by Ulrich and Thiele, the Solow Polasky Diversity (SP Diversity, also known as Magnitude [1]) served as a metric, other diversity indicators could be considered. We examine the parameter-free Max-Min Diversity and the Riesz $$s$$ s -Energy, which features uniformly distributed solution sets. Focusing on multi-objective diversity optimization, we discuss different diversity indicators from the perspective of indicator-based evolutionary algorithms with multiple objectives. We examine theoretical, computational, and practical properties of these indicators, such as monotonicity in species, twinning, monotonicity in distance, strict monotonicity in distance, uniformity of maximizing point sets, computational effort for a set of size $$n$$ n , single-point contributions, subset selection, and submodularity. We present new theorems—including a proof of the NP-hardness of the Riesz $$s$$ s -Energy Subset Selection Problem—and consolidate existing results from the literature. In the experiments, we apply these indicators in the NOAH algorithm to analyze search dynamics via an example. We study how optimizing one indicator impacts others and propose NOAH-specific modifications for the Max-Min indicator.

Even with the recent theoretical advancements that dramatically reduced the complexity of Integer Programming (IP), heuristics remain the dominant problem-solvers for this difficult category. This study seeks to establish the foundation for Evolution Strategies (ESs), a class of randomized search heuristics inherently designed for continuous spaces. We particularly focus on ESs for their intrinsic mixed-integer capabilities, well-developed self-adaptation mechanisms, and high efficacy in handling unbounded search spaces. ESs already excel in treating IP in practice, but accomplish it via discretization and by applying sophisticated patches to their continuous operators, while persistently using the ℓ2-norm as their operation pillar. We lay foundations for discrete search by adopting the ℓ1 -norm, accounting for the suitable step-size, and exploring mutation distributions for unbounded integer decision variables. We narrow down to the Truncated Normal (TN) and Double Geometric (DG) distributions, explore their theoretical properties, including entropy functions, and propose a procedure to generate scalable correlated mutations. Our investigations are accompanied by extensive numerical simulations, which consistently support the claim that the DG distribution is better suited for unbounded integer search. We link our theoretical perspective to empirical evidence indicating that an ES with correlated DG mutations outperforms other strategies over non-separable quadratic IP. We conclude that substituting the default TN distribution with the DG is theoretically justified and practically beneficial. We also advocate for the use of the ℓ1-norm over the ℓ2-norm based on the empirical indications presented, although it remains to be formally proven and/or rigorously benchmarked.

Within the current literature on multi-objective optimization, there is a scarcity of comparisons between equation-based white-box solvers to evolutionary black-box solvers. It is commonly held that when dealing with linear and quadratic models, equation-based deterministic solvers are generally the preferred choice. The present study aims at challenging this hypothesis, and we show that particularly in box-constrained mixed-integer (MI) problems it is worth employing evolutionary methods when the goal is to achieve a good approximation of a Pareto frontier. To do so, this paper compares a mathematical programming approach with an evolutionary method for set-oriented Pareto front approximation of bi-objective quadratic MI optimization problems. The focus is on convex quadratic under-constrained models wherein the decision variables are either tightly or loosely bounded by box-constraints. Through an empirical assessment of families of quadratic models across varying Hessian forms, variable ranges, and condition numbers, the study compares the performance of the CPLEX-based Diversity Maximization Approach to a state-of-the-art evolutionary multi-objective optimization meta-heuristic with MI mutation and crossover operators. We identify and explain strengths and weaknesses of both approaches when dealing with loosely bounded box-constraints, and prove a theorem regarding the potential undecidability of such multi-objective problems featuring unbounded integer decision variables. The empirical results systematically confirm that black-box and white-box solvers can be competitive, especially in the case of loose box-constraints.

Various interactive evolutionary multiobjective optimization methods have been proposed in the literature for problems with multiple, conflicting objective functions. In these methods, a decision maker, who is a domain expert, iteratively provides preference information to guide the solution process while gaining insight into the problem. To compare interactive evolutionary multiobjective optimization methods, a preference-based hypervolume indicator (PHI) has been proposed to quantify the performance of the methods. PHI was the first indicator designed based on some desirable properties of indicators for interactive evolutionary multiobjective optimization methods. However, it has some shortcomings, such as excluding some potentially interesting solutions and being limited to consider a reference point as a type of preference information. In this paper, a modified indicator called PHI+ is proposed to address the mentioned drawbacks. PHI+ modifies the region of interest in PHI. While PHI is directed at methods where a decision maker provides preference information in the form of a reference point, PHI+ is applicable for methods that utilize desirable ranges of objective function values as preference information. Therefore, PHI+ is the first indicator that can handle preference information provided as desirable ranges when evaluating interactive methods. Experimental results show that PHI+ can also better distinguish differences in the performance of interactive evolutionary multiobjective optimization methods.

In recent years, interactive evolutionary multiobjective optimization methods have been getting more and more attention. In these methods, a decision maker, who is a domain expert, is iteratively involved in the solution process and guides the solution process toward her/his desired region with preference information. However, there have not been many studies regarding the performance evaluation of interactive evolutionary methods. On the other hand, indicators have been developed for a priori methods, where the DM provides preference information before optimization. In the literature, some studies treat interactive evolutionary methods as a series of a priori steps when assessing and comparing them. In such settings, indicators designed for a priori methods can be utilized. In this paper, we propose a novel performance indicator for interactive evolutionary multiobjective optimization methods and show how it can assess the performance of these interactive methods as a whole process and not as a series of separate steps. In addition, we demonstrate the shortcomings of using indicators designed for a priori methods for comparing interactive evolutionary methods.

Abstract A building spatial design (BSD) determines external and internal walls and ceilings of a building. The design space has a hierarchical structure, in which decisions on the existence or non-existence of spatial components determine the existence of variables related to these spaces, such as sizing and angles. In the optimization of BSDs it is envisioned to optimize various performance indicators from multiple disciplines in concert, such as structural, functional, thermal, and daylight performance. Existing representations of design spaces suffer from severe limitations, such as only representing orthogonal designs or representing the structures in parametric superstructure, allowing only for limited design variations. This paper proposes prism nets - a new way of representing the search space of BSDs based on triangulations defining space filling collections of triangular prisms that can be combined via coloring parameters to spaces. Prism nets can accommodate for non-orthogonal designs and are flexible in terms of topological variations. We follow the guidelines for representation and operator design proposed in the framework of metric-based evolutionary algorithms. The main contribution of the paper is a detailed discussion of the search space representation and corresponding mutation operators. Moreover, a proof of concept example demonstrates the integration into multi-objective evolutionary algorithms and provides first results on a simple, but reproducible, benchmark problem.

Abstract Optimization problems with multiple objectives and many input variables inherit challenges from both large-scale optimization and multi-objective optimization. To solve the problems, decomposition and transformation methods are frequently used. In this study, an improved control variable analysis is proposed based on dominance and diversity in Pareto optimization. Further, the decomposition method is used in a cooperative coevolution framework with orthogonal sampling mutation. The algorithm’s performances are compared against the weighted optimization framework. The results show that the proposed decomposition method has much better accuracy compared to the traditional method. The results also show that the cooperative coevolution framework with a good grouping is very competitive. Additionally, the number of search directions in orthogonal sampling can be easily configured. A small number of search directions will reduce the search space greatly while also restricting the area that can be explored and vice versa.

The most relevant property that a quality indicator (QI) is expected to have is Pareto compliance, which means that every time an approximation set strictly dominates another in a Pareto sense, the indicator must reflect this. The hypervolume indicator and its variants are the only unary QIs known to be Pareto-compliant but there are many commonly used weakly Pareto-compliant indicators such as R2, IGD+, and ε+. Currently, an open research area is related to finding new Pareto-compliant indicators whose preferences are different from those of the hypervolume indicator. In this article, we propose a theoretical basis to combine existing weakly Pareto-compliant indicators with at least one being Pareto-compliant, such that the resulting combined indicator is Pareto-compliant as well. Most importantly, we show that the combination of Pareto-compliant QIs with weakly Pareto-compliant indicators leads to indicators that inherit properties of the weakly compliant indicators in terms of optimal point distributions. The consequences of these new combined indicators are threefold: (1) to increase the variety of available Pareto-compliant QIs by correcting weakly Pareto-compliant indicators, (2) to introduce a general framework for the combination of QIs, and (3) to generate new selection mechanisms for multiobjective evolutionary algorithms where it is possible to achieve/adjust desired distributions on the Pareto front.

Abstract We introduce novel concepts to solve multiobjective optimization problems involving (computationally) expensive function evaluations and propose a new interactive method called O-NAUTILUS. It combines ideas of trade-off free search and navigation (where a decision maker sees changes in objective function values in real time) and extends the NAUTILUS Navigator method to surrogate-assisted optimization. Importantly, it utilizes uncertainty quantification from surrogate models like Kriging or properties like Lipschitz continuity to approximate a so-called optimistic Pareto optimal set. This enables the decision maker to search in unexplored parts of the Pareto optimal set and requires a small amount of expensive function evaluations. We share the implementation of O-NAUTILUS as open source code. Thanks to its graphical user interface, a decision maker can see in real time how the preferences provided affect the direction of the search. We demonstrate the potential and benefits of O-NAUTILUS with a problem related to the design of vehicles.

In drug discovery, classification is a well established in silico method based on machine learning algorithms. However, since the activity value on a target protein is described as a continuous value, a regressional approach is worth being considered as well. The results of the regression can then be turned into classification results with a given threshold value. To further improve the results, a new method called optimally weighted ensembles that uses a combination of more than one model to build a better performing ensemble of models, is applied here. This method is used in the context of drug discovery for the first time. Naturally, it is crucial to choose suitable models for the specific problem, if any prior knowledge is available. Statistical significance of the approach is verified using a second dataset with a different target. In this work, we show to what extent the obtained classification results compare to previous, highly optimized, single model results as presented in previous work. Furthermore, the results are compared to ensembles where none of the contributing models were optimized beforehand. All case studies are performed using the publicly available database ChEMBL1.

For almost 20 years, quality indicators (QIs) have promoted the design of new selection mechanisms of multiobjective evolutionary algorithms (MOEAs). Each indicator-based MOEA (IB-MOEA) has specific search preferences related to its baseline QI, producing Pareto front approximations with different properties. In consequence, an IB-MOEA based on a single QI has a limited scope of multiobjective optimization problems (MOPs) in which it is expected to have a good performance. This issue is emphasized when the associated Pareto front geometries are highly irregular. In order to overcome these issues, we propose here an island-based multiindicator algorithm (IMIA) that takes advantage of the search biases of multiple IB-MOEAs through a cooperative scheme. Our experimental results show that the cooperation of multiple IB-MOEAs allows IMIA to perform more robustly (considering several QIs) than the panmictic versions of its baseline IB-MOEAs as well as several state-of-the-art MOEAs. Additionally, IMIA shows a Pareto-front-shape invariance property, which makes it a remarkable optimizer when tackling MOPs with complex Pareto front geometries.

The Cooperative Coevolution (CC) framework is the state of the art for solving large scale global optimization (LSGO) problems. A particular challenge in using CC lies in the decomposition of variables and resource allocation. In this work, the decomposition phase of the framework is performed in two stages to address both variable interaction and efficient resource allocation. The algorithm starts with differential analysis followed by differential grouping. The differential analysis allows efficient resource allocation while the differential grouping will detect variable interactions. The differential grouping will act on a small number of variables and will not consume as much computational budget as a single-stage grouping. While not all variable interactions will be detected, separable variables will be recognized hence specialized solvers for separable problems can be employed on these subproblems. In this work, the twostage grouping CC (TSCC) is paired with a hybrid algorithm where sep-CMA-ES is used to solve the separable subproblem and CMA-ES is used to solve the non-separable subproblems, the algorithm is referred as TSCC-CMAES. A comparison between TSCC and CC with groups based on either differential analysis or differential grouping is carried out. In general, TSCC could outperform the two single-stage grouping methods. Additionally, the TSCC-CMAES shows a competitive advantage on a number of more complex problems against state-of-the-art algorithms and it is shown that the effect of the population size and group size is crucial in achieving these results.

Abstract KrigingorGaussian Process Regressionis applied in many fields as a non-linear regression model as well as a surrogate model in the field of evolutionary computation. However, the computational and space complexity of Kriging, that is cubic and quadratic in the number of data points respectively, becomes a major bottleneck with more and more data available nowadays. In this paper, we propose a general methodology for the complexity reduction, called cluster Kriging, where the whole data set is partitioned into smaller clusters and multiple Kriging models are built on top of them. In addition, four Kriging approximation algorithms are proposed as candidate algorithms within the new framework. Each of these algorithms can be applied to much larger data sets while maintaining the advantages and power of Kriging. The proposed algorithms are explained in detail and compared empirically against a broad set of existing state-of-the-art Kriging approximation methods on a well-defined testing framework. According to the empirical study, the proposed algorithms consistently outperform the existing algorithms. Moreover, some practical suggestions are provided for using the proposed algorithms.

In this paper, we propagate the use of a set-based Newton method that enables computing a finite size approximation of the Pareto front (PF) of a given twice continuously differentiable bi-objective optimization problem (BOP). To this end, we first derive analytically the Hessian matrix of the hypervolume indicator, a widely used performance indicator for PF approximation sets. Based on this, we propose the hypervolume Newton method (HNM) for hypervolume maximization of a given set of candidate solutions. We first address unconstrained BOPs and focus further on first attempts for the treatment of inequality constrained problems. The resulting method may even converge quadratically to the optimal solution, however, this property is-as for all Newton methods-of local nature. We hence propose as a next step a hybrid of HNM and an evolutionary strategy in order to obtain a fast and reliable algorithm for the treatment of such problems. The strengths of both HNM and hybrid are tested on several benchmark problems and comparisons of the hybrid to state-of-the-art evolutionary algorithms for hypervolume maximization are presented.

In recent years, several indicator-based multi-objective evolutionary algorithms (IB-MOEAs) have been proposed. Each IB-MOEA presents different search preferences depending on the quality indicator (QI) that it uses in its selection mechanism. However, due to these search biases, IB-MOEAs behave differently on each multi-objective optimization problem, producing Pareto front approximations whose characteristics are related to the QI on which they are based. In this paper, we propose a novel algorithm based on the island model that aims to take advantage of the cooperation of individual IB-MOEAs based on the indicators hypervolume, R2, IGD+,+, and Δpwith the aim of improving both convergence and distribution of the Pareto fronts produced. Our experimental results, taking into account seven quality indicators, empirically show that the cooperation of several IB-MOEAs is better than using panmictic versions of them. Additionally, we also show that the performance of our proposal does not depend on the Pareto front shape of the problem being solved.

In this paper, a new real-world application problem, i.e., the vehicle fleet maintenance scheduling optimization problem, is defined and a specialized multi-objective evolutionary algorithm framework (grouping strategy, three vector chromosome and corresponding genetic operators) is proposed to solve the problem. State-of-the-art multi-objective evolutionary algorithms such as NSGA-III, SMS-EMOA, DI-MOEA are employed in the proposed algorithm framework to solve the problem, and their behavior is investigated. Although DI-MOEA is used the first time for a real-world application problem, its performance is better than other algorithms for some instances.

Classification based on machine learning algorithms is a widely used technique in contemporary in silico methods for drug discovery. However, typically the performance of the classification tool is evaluated based on a scalar performance score and essential information, such as the balance between false positive rates (FPRs)and false negative rates (FNRs)is not directly assessed. Moreover, there might be a large number of molecular features that are not relevant for the classification task and merely slow down the computations or add noise to the learning process. In this paper we adopt an approach that previously was used for the classification of text messages (spam/no-spam)to the classification of drug compounds (active/inactive). By considering the minimization of the classification costs (FPR, FNR)and the minimization of the number of features as separate optimization tasks, we demonstrate that it is possible to develop a more informative and versatile tool for drug discovery. We show, how to derive and evaluate 2-D and 3-D Pareto fronts for the classification of small compounds in active and non-active (similar studies could be conducted for toxic/non-toxic classification, and on other chemically relevant properties). We demonstrate the applicability of the method on a small data set for bio-activity prediction of ligands.

Effective soil-sampling is essential for the construction of prescription maps used in Precision Agriculture for Variable Rate Application of nutrients. In practice, designing a field sampling plan is subject to hard limitations, merely due to the associated expenses, where only a few sample points are taken for evaluation. The accuracy of constructed maps is affected by the number of sampling points, their geographical dispersion and their coverage of the feature space. To improve the accuracy, ancillary data in the form of low-cost, high-resolution field scans could be used for inferring statistical measures for devising the high-cost sampling plan. The current study targets algorithmically-guided sampling plans using available ancillary data. We propose possible models for quantifying spatial coverage and diversity concerning the ancillary data. We investigate models as objective functions, devise Pareto optimization problems and solve them using NSGA-II. We analyzed the obtained sampling plans in an agricultural field, and suggest statistical tools for sample-size determination and plans' ranking according to additional information criteria. We argue that our approach is successful in attaining a practical sampling plan, constituting a fine trade-off between objectives, and possessing no discrepancies.

The technique of parallelization is a trend in the field of Bayesian global optimization (BGO) and is important for real-world applications because it can make full use of CPUs and speed up the execution times. This paper proposes a multi-point mechanism of the expected hypervolume improvement (EHVI) for multi-objective BGO (MOBGO) by the utilization of the truncated EHVI (TEHVI). The basic idea is to divide the objective space into several sub-objective spaces and then search for the optimal solutions in each sub-objective space by using the TEHVI as the infill criterion. We studied the performance of the proposed algorithm and performed comparisons with Kriging believer technique (KB) on five scientific benchmarks and a real-world application problem (i.e., a low-fidelity multi-objective airfoil optimization design). The stochastic experimental results show that the proposed algorithm performs better than the KB with respect to the hypervolume indicator, indicating that the proposed method provides an efficient parallelization technique for MOBGO.

We continue recent work on the definition of multimodality in multiobjective optimization (MO) and the introduction of a test bed for multimodal MO problems. This goes beyond well-known diversity maintenance approaches but instead focuses on the landscape topology induced by the objective functions. More general multimodal MO problems are considered by allowing ellipsoid contours for single-objective subproblems. An experimental analysis compares two MO algorithms, one that explicitly relies on hypervolume gradient approximation, and one that is based on local search, both on a selection of generated example problems. We do not focus on performance but on the interaction induced by the problems and algorithms, which can be described by means of specific characteristics explicitly designed for the multimodal MO setting. Furthermore, we widen the scope of our analysis by additionally applying visualization techniques in the decision space. This strengthens and extends the foundations for Exploratory Landscape Analysis (ELA) in MO.

Generating more evenly distributed samples in high dimensional search spaces is the major purpose of the recently proposed mirrored sampling technique for evolution strategies. The diversity of the mutation samples is enlarged and the convergence rate is therefore improved by the mirrored sampling. Motivated by the mirrored sampling technique, this article introduces a new derandomized sampling technique called mirrored orthogonal sampling. The performance of this new technique is both theoretically analyzed and empirically studied on the sphere function. In particular, the mirrored orthogonal sampling technique is applied to the well-known Covariance Matrix Adaptation Evolution Strategy (CMA-ES). The resulting algorithm is experimentally tested on the well-known Black-Box Optimization Benchmark (BBOB). By comparing the results from the benchmark, mirrored orthogonal sampling is found to outperform both the standard CMA-ES and its variant using mirrored sampling.

In the field of multi-objective optimization algorithms, multi-objective Bayesian Global Optimization (MOBGO) is an important branch, in addition to evolutionary multi-objective optimization algorithms. MOBGO utilizes Gaussian Process models learned from previous objective function evaluations to decide the next evaluation site by maximizing or minimizing an infill criterion. A commonly used criterion in MOBGO is the Expected Hypervolume Improvement (EHVI), which shows a good performance on a wide range of problems, with respect to exploration and exploitation. However, so far, it has been a challenge to calculate exact EHVI values efficiently. This paper proposes an efficient algorithm for the exact calculation of the EHVI for in a generic case. This efficient algorithm is based on partitioning the integration volume into a set of axis-parallel slices. Theoretically, the upper bound time complexities can be improved from previously $$O (n^2)$$ and $$O(n^3)$$ , for two- and three-objective problems respectively, to $$\varTheta (n\log n)$$ , which is asymptotically optimal. This article generalizes the scheme in higher dimensional cases by utilizing a new hyperbox decomposition technique, which is proposed by Dächert et al. (Eur J Oper Res 260(3):841–855, 2017). It also utilizes a generalization of the multilayered integration scheme that scales linearly in the number of hyperboxes of the decomposition. The speed comparison shows that the proposed algorithm in this paper significantly reduces computation time. Finally, this decomposition technique is applied in the calculation of the Probability of Improvement (PoI).

Bayesian Global Optimization algorithm is designed to optimize expensive objective functions with small evaluation budget. This algorithm employs a surrogate model and assesses the potential improvement of unseen solutions through the so-called infill-criterion. A novel infill-criterion proposed in our previous work is derived from the moment-generating function of the improvement. In contrast to other techniques, it features a continuous parameter that can be used to adjust the exploration-exploitation tradeoff smoothly. In this work, two cooling strategies (linear and exponential) are adopted to enhance the explorative behavior in the early stage of the search and the exploitative effect in the final converging stage. Moreover, the initial temperature and cooling speed are investigated on some selected multi-modal functions, showing that the good setting of those two parameters depends on the problems specifics. The proposed Bayesian optimization with cooling strategy is tested on well-known BBOB benchmark. The results shows that without tuning the initial temperature and cooling speed, the proposed approach improves the performance on a range of multi-modal functions as compared to the commonly used expected improvement criterion.

Multi-objective optimization in aerodynamic design plays an important role in capturing the trade-off and useful knowledge that would be useful for real-world design processes. In the preliminary design phase, aerodynamic designers usually have an interest in focusing the optimization process in a certain direction of interest. To this end, we propose the use of user preference multi-objective Bayesian global optimization (MOBGO) for aerodynamic design using truncated expected hypervolume improvement (TEHVI). Taking into account the apriori knowledge of objective functions, TEHVI acts as an infill criterion to search for the optimal solutions based on the Kriging models in MOBGO. In TEHVI-MOBGO, the first step is to obtain a coarse approximation of the Pareto front in order to capture the general trend and trade off using standard EHVI; following this step, TEHVI is then applied to focus the search on a defined region of interest. We demonstrate the capabilities and usefulness of TEHVI method on the design optimization of an inviscid transonic wing and a viscous transonic airfoil in order to minimize the drag coefficient and absolute value of pitching moment, which leads to a reduced fuel burn and easier control characteristic.

In almost no other field of computer science, the idea of using bio-inspired search paradigms has been so useful as in solving multiobjective optimization problems. The idea of using a population of search agents that collectively approximate the Pareto front resonates well with processes in natural evolution, immune systems, and swarm intelligence. Methods such as NSGA-II, SPEA2, SMS-EMOA, MOPSO, and MOEA/D became standard solvers when it comes to solving multiobjective optimization problems. This tutorial will review some of the most important fundamentals in multiobjective optimization and then introduce representative algorithms, illustrate their working principles, and discuss their application scope. In addition, the tutorial will discuss statistical performance assessment. Finally, it highlights recent important trends and closely related research fields. The tutorial is intended for readers, who want to acquire basic knowledge on the mathematical foundations of multiobjective optimization and state-of-the-art methods in evolutionary multiobjective optimization. The aim is to provide a starting point for researching in this active area, and it should also help the advanced reader to identify open research topics.

In this paper solution approaches for solving the building spatial design optimisation problem for structural and energy performance are advanced on multiple fronts. A new initialisation operator is introduced to generate an unbiased initial population for a tailored version of SMS-EMOA with problem specific operators. Improvements to the mutation operator are proposed to eliminate bias and allow mutations consisting of multiple steps. Moreover, landscape analysis is applied in order to explore the landscape of both objectives and investigate the behaviour of the mutation operator. Parameter tuning is applied with the irace package and the Mixed Integer Evolution Strategy to find improved parameter settings and explore tuning with a relatively small number of expensive evaluations. Finally, the performances of the standard and tailored SMS-EMOA algorithms with tuned parameters are compared.

This paper investigates Earth observation scheduling of agile satellite constellation based on evolutionary multiobjective optimization (EMO). The mission planning of agile Earth observation satellite (AEOS) is to select and specify the observation activities to acquire images on the earth surface. This should be done in accordance with operational constraints and in order to maximize certain objectives. In this paper three objectives are considered, i. e. profit, quality and timeliness. Preference-based EMO methods are introduced to generate solutions preferable for the decision maker, whose preference is expressed by a reference point. An improved algorithm based on R-NSGA-II, which is named CD-NSGA-II, is proposed and compared with other algorithms in various scheduling scenarios. Results show that the chosen preference modeling paradigm allows to focus search on the interesting part of the Pareto front. Moreover, the tested algorithms behave differently with the change of reference point and degree of conflicts, among them CD-NSGA-II has the best overall performance. Suggestions on applying these algorithms in practice are also given in this paper.

In this paper, a target region based multiobjective evolutionary algorithm framework is proposed to incorporate preference into the optimization process. It aims at finding a more fine-grained resolution of a target region without exploring the whole set of Pareto optimal solutions. It can guide the search towards the regions on the Pareto Front which are of real interest to the decision maker. The algorithm framework has been combined with SMS-EMOA, R2-EMOA, NSGA-II to form three target region based multiobjective evolutionary algorithms: T-SMS-EMOA, T-R2-EMOA and T-NSGA-II. In these algorithms, three ranking criteria are applied to achieve a well-converged and well-distributed set of Pareto optimal solutions in the target region. The three criteria are: 1. Non-dominated sorting; 2. indicators (hypervolume or R2 indicator) or crowding distance in the new coordinate space (i.e. target region) after coordinate transformation; 3. the Chebyshev distance to the target region. Rectangular and spherical target regions have been tested on some benchmark problems, including continuous problems and discrete problems. Experimental results show that new algorithms can handle the preference information very well and find an adequate set of Pareto-optimal solutions in the preferred regions quickly. Moreover, the proposed algorithms have been enhanced to support multiple target regions and preference information based on a target point or multiple target points. Some results of enhanced algorithms are presented.

Network centrality plays an important role in network analysis - especially in social and economic network analysis such as identification of the most popular actor and artist in the Hollywood community, or to find the most influential scientist in a citation network, or politician in democratic elections. Furthermore, finding an important player for the growth of economics in a region can be important to improve future welfare, or to find important hubs for spreading an important message in crisis management. Many algorithms have been proposed to identify a set of key players in a single network. But in the real world with more complicated data sets we need not only to identify a single player but a set of key players. Moreover, we may have to use different types of links simultaneously, e.g., different social networks, in order to define how influential a node is. This situation can be modelled by multiplex network data. For a multiplex network the set of nodes stays the same, while there are multiple sets of edges. The utilization of such information can be viewed as a multiple objective decision analysis problem. In this paper, we propose a new approach in identifying a network centrality based on a many-objective optimization approach, where the nodes are the potential points to be selected and the objectives are their centrality in the different layers of the network. This yields a new approach to analyse network centrality in multiplex network. For this approach, we propose to compute the Pareto fronts of network centrality of nodes, where maximization of centrality in layer defines its own objective. As a case study, we compute the Pareto fronts for model problems with artificial network and real networks for economic data sets to show on how to find the network centrality trade-offs between different layers and identify efficient sets of key nodes.

Let $B$ be a set of $n$ axis-parallel boxes in $\mathbb{R}^d$ such that each box has a corner at the origin and the other corner in the positive quadrant of $\mathbb{R}^d$, and let $k$ be a positive integer. We study the problem of selecting $k$ boxes in $B$ that maximize the volume of the union of the selected boxes. This research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known running time in any dimension $d \ge 3$ is $Ω\big(\binom{n}{k}\big)$. We show that: - The problem is NP-hard already in 3 dimensions. - In 3 dimensions, we break the bound $Ω\big(\binom{n}{k}\big)$, by providing an $n^{O(\sqrt{k})}$ algorithm. - For any constant dimension $d$, we present an efficient polynomial-time approximation scheme.

Efficient Global Optimization (EGO) is an effective method to optimize expensive black-box functions and utilizes Kriging models (or Gaussian process regression) trained on a relatively small design data set. In real-world applications, such as experimental optimization, where a large data set is available, the EGO algorithm becomes computationally infeasible due to the time and space complexity of Kriging. Recently, the so-called Cluster Kriging methods have been proposed to reduce such complexities for the big data, where data sets are clustered and Kriging models are built on each cluster. Furthermore, Kriging models are combined in an optimal way for the prediction. In addition, we analyze the Cluster Kriging landscape to adopt the existing infill-criteria, e.g., the expected improvement. The approach is tested on selected global optimization problems. It is shown by the empirical studies that this approach significantly reduces the CPU time of the EGO algorithm while maintaining the convergence rate of the algorithm.

Bayesian Optimization or Efficient Global Optimization (EGO) is a global search strategy that is designed for expensive black-box functions. In this algorithm, a statistical model (usually the Gaussian process model) is constructed on some initial data samples. The global optimum is approached by iteratively maximizing a so-called acquisition function, that balances the exploration and exploitation effect of the search. The performance of such an algorithm is largely affected by the choice of the acquisition function. Inspired by the usage of higher moments from the Gaussian process model, it is proposed to construct a novel acquisition function based on the moment-generating function (MGF) of the improvement, which is the stochastic gain over the current best fitness value by sampling at an unknown point. This MGF-based acquisition function takes all the higher moments into account and introduces an additional real-valued parameter to control the trade-off between exploration and exploitation. The motivation, rationale and closed-form expression of the proposed function are discussed in detail. In addition, we also illustrate its advantage over other acquisition functions, especially the so-called generalized expected improvement.

The vehicle routing problem is a classical combinatorial optimization problem. This work is about a variant of the vehicle routing problem with dynamically changing orders and time windows. In real-world applications often the demands change during operation time. New orders occur and others are canceled. In this case new schedules need to be generated on-the-fly. Online optimization algorithms for dynamical vehicle routing address this problem but so far they do not consider time windows. Moreover, to match the scenarios found in real-world problems adaptations of benchmarks are required. In this paper, a practical problem is modeled based on the procedure of daily routing of a delivery company. New orders by customers are introduced dynamically during the working day and need to be integrated into the schedule. A multiple ant colony algorithm combined with powerful local search procedures is proposed to solve the dynamic vehicle routing problem with time windows. The performance is tested on a new benchmark based on simulations of a working day. The problems are taken from Solomon's benchmarks but a certain percentage of the orders are only revealed to the algorithm during operation time. Different versions of the MACS algorithm are tested and a high performing variant is identified. Finally, the algorithm is tested in situ: In a field study, the algorithm schedules a fleet of cars for a surveillance company. We compare the performance of the algorithm to that of the procedure used by the company and we summarize insights gained from the implementation of the real-world study. The results show that the multiple ant colony algorithm can get a much better solution on the academic benchmark problem and also can be integrated in a real-world environment.

Kriging-based Global optimization has been proposed and extensively used for solving black-box optimization problems with expensive function evaluations. The performance of such algorithm relies heavily on the effectiveness of the infill criterion that is used to decide which point to evaluate next. Two common infill criteria are, the probability of improvement (PI) and the expected improvement (EI). The PI results in solutions that have a low risk of failure, that is the solution is not improved in the next round. However the expected gain can be small. EI has a higher risk of failure but maximizes expected gain. This paper views the maximization of PI and EI as a bi-objective optimization problem and suggest a fast and precise gradient-based hypervolume ascent algorithm to compute the Pareto front. The computed Pareto front can be beneficial in different scenarios: Firstly, it can be used for decision making in experimental optimization, when a human decision maker has to decide between a `low risk, low gain' or a `high risk, high gain' strategy. Another application is to use different points on the Pareto front in a multi-point efficient global optimization strategy to balance between exploitation and exploration. The algorithm is validated on two objective functions and compared to other well-known multi-objective optimization algorithms. As a side result we also analyze the landscape of the PI and EI infill criterion and provide closed-form gradient expressions of them.

In optimization with expensive black box evaluations, the expected improvement algorithm (also called efficient global optimization) is a commonly applied method. It uses Gaussian Processes (or Kriging) to build a model of the objective function and uses the expected improvement as an infill criterion, taking into account both — predictive mean and variance. It has been generalized to multi-objective optimization using the expected hypervolume improvement, which measures the expected gain in the hypervolume indicator of a Pareto front approximation. However, this criterion assumes an unbounded objective space even if it is often known a-priori that the objective function values are within a prescribed range, e.g., lower bounded by zero. To take advantage of such a-priori knowledge, this paper introduces the truncated expected hypervolume improvement and a multiobjective efficient global optimization method that is based on it. In this paper it is shown how to compute the truncated expected hypervolume improvement exactly and efficiently. Then it is tested as an infill criterion in efficient global optimization. It is shown that it can effectively make use of a-priori knowledge and achieve better results in cases where such knowledge is given. The usefulness of the new approach is demonstrated in benchmark examples and applications from robust PID (proportionalintegral-derivative) controller optimization. The empirical studies in this paper are confined to the bi-objective case.

Kriging or Gaussian Process Regression has been successfully applied in many fields. One of the major bottlenecks of Kriging is the complexity in both processing time (cubic) and memory (quadratic) in the number of data points. To overcome these limitations, a variety of approximation algorithms have been proposed. One of these approximation algorithms is Optimally Weighted Cluster Kriging (OWCK). In this paper, OWCK is extended and enhanced by the use of fuzzy clustering methods in order to increase the accuracy. Several options are proposed and evaluated against both the original OWCK and a variety of other Kriging approximation algorithms.

In this contribution a derivative-free memetic algorithm (MA) for the design optimization of chemical processes is introduced. Design optimization problems are characterized by nonlinear cost functions and highly constrained and multi-modal search spaces. The MA is a combination of an evolution strategy that addresses the global optimization of discrete and continuous design decisions and a derivative-free optimization method (DFO) that performs a local optimization of the continuous sub-problems that remain after fixing the discrete decisions. The MA calls a process simulation software to simulate the design alternatives, i.e. the evaluation of the objective and the constraints is a black box. In this contribution, the focus lies on the selection of a suitable DFO solver for efficiently solving the continuous constrained sub-problems. Based on latin-hypercube samplings of sub-problems of two instances of a real-world case study, surrogate models for the objective and the constraints were generated. A set of DFO methods was tested and compared on the surrogate models. The method that showed the best performance was coupled to the MA which was then applied to the real-world case study.

Truck Loading Optimization problems from practice tend to involve a lot of constraints. In automatically solving these problems, a rule set has to be compiled that governs the generation of valid solutions. This rule set is either defined manually, or distilled automatically by analyzing solutions of human planners. This paper describes the extension of an existing optimization approach with statistics from man-made solutions through a so-called informed mutation operator. Solutions are scored on quality and on the percentage of violation, i.e., a combination of stacked boxes that did not occur in a separate set of man-made solutions. Applying the informed mutation operator decreases the average violation percentage in solutions, as well as achieving improved solution quality through fitting more boxes into a container.

Multi-objective optimization problems with many objective functions (3> 3) are difficult to solve. This is because the time complexity of optimization algorithms often grows fast with the number of objective functions and the results of many objective optimization algorithms (Pareto front approximation) require large memory and their interpretation can be difficult for the decision maker. It is therefore attractive to reduce complexity of these problems. This can be achieved by decomposing them into a set of independent lower dimensional subproblems, or by aggregating some objective functions into a single objective function. This work introduces a new approach for decomposition and aggregation based on techniques from social network analysis. The key idea is to interpret an objective function as a node (agent) in a social network, and arcs between nodes indicate relationships: Negatively weighted arcs stand for conflicting objectives, zero weighted arcs for independent objectives, and positively weighted arcs for objectives that support each other. Using well-known algorithms BGLL for community detection we show that — given certain preconditions — it is possible to decompose a many objective optimization problem to a set of lower dimensional multi-objective optimization problems. This makes it easier to solve the problem and interpret the resulting trade-off (hyper-)surfaces. In the paper, after introducing the idea of Communtiy Detection for Many-Objective Optimization (CoDeMO), we lay out an interactive workflow and test it on simple, scalable many-objective optimization problems-variants of facility location problems — and thereby provide a prove-of-concept study.

Most research into the Container Loading problem has been done on theoretical problem sets and while taking one or two constraints into account. In this paper we discuss the successful implementation of a self-adaptive Genetic Algorithm applying only mutation, with a variable mutation rate. This is applied to a real-world problem with actual problem instances from industry. We introduce an abstract, indirect representation for the considered loadings together with two mutation strategies. Solutions of these different strategies are compared with each other, a static mutation rate GA, and with solutions created by human planners as used in industry, for a set of over 500 realworld problem instances. Furthermore, we examine how our automated results compare to those generated by experienced human planners, showing that they are valid loadings and match fitness values.

In this paper, we propose an ant based solver to solve dynamic vehicle routing problem with time windows and multiple priorities (DVRPTWMP). In this problem, a fleet of vehicles will service a number of customers with time windows. But part of customers are unknown and revealed dynamically during the execution of the routes. More specifically, customers have different priority levels. Customers with higher priority should have a higher quality of service. The quality of service is measured by the sum of the expected delay time between the arrival time and the earliest available beginning service time of all customers. The goal is to minimize the traveling distance while minimizing the total delay time of customers. First, a new benchmark is generated for DVRPTWMP based on van Veen's benchmark which is a dynamical extension of Solomon's 100 customers benchmark. Then, the ant based solver is introduced and two strategies based on ant colony algorithm are proposed to deal with priorities of customers. One is servicing high priority level customers immediately using the nearest vehicle. The other is giving a penalty to the delay time, combining the penalty and traveling distance and minimizing them together. Finally, the results show that the second strategy performs better than the first one.

This paper presents and analyses an engineered expected hypervolume improvement (EHVI) algorithm for solving the problem of PID parameter tuning and the optimization problem of controlling the substrate feed of a biogas plant. The EHVI is the expected value of the increment of the hypervolume indicator given a Pareto front approximation and a predictive multivariate Gaussian distribution of a new point. To solve this problem, S-metric selection-based efficient global optimization (SMS-EGO), EHVI based efficient global optimization (EHVIEGO) and SMS-EMOA are used and compared in both the PID parameter tuning problem and for biogas plant feed optimization. The results of the experiments show that surrogate model based algorithms perform better than SMS-EMOA, and the performance of EHVI-EGO is slightly better than SMS-EGO.

Complex networks rely on their structural robustness for their function and performance. Considering that redundancy backup is frequently used to enhance the robustness of complex networks, we try to find better redundancy backup strategy by using optimization methods. In this paper, our contributions are twofold. First, we prove that natural connectivity is suitable for measuring network robustness. Second, a robustness optimization algorithm is proposed based on GA, while it is different from traditional GA. The method of coding, crossover and mutation operations are all improved in this research. Extensive experiments on real-world datasets demonstrate that the effectiveness of our methods is better than the classical rich-rich redundancy backup strategy.

The receiver operating characteristic (ROC) is commonly used to analyze the performance of classifiers in data mining. An important topic in ROC analysis is the ROC convex hull (ROCCH), which is the least convex majorant (LCM) of the empirical ROC curve and covers potential optima for a given set of classifiers. ROCCH maximization problems have been taken as multiobjective optimization problem (MOPs) in some previous work. However, the special characteristics of ROCCH maximization problem makes it different from traditional MOPs. In this paper, the difference will be discussed in detail and a new convex hull-based multiobjective genetic programming (CH-MOGP) is proposed to solve ROCCH maximization problems. Specifically, convex hull-based without redundancy sorting (CWR-sorting) is introduced, which is an indicator-based selection scheme that aims to maximize the area under the convex hull. A novel selection procedure is also proposed based on the proposed sorting scheme. It is hypothesized that by using a tailored indicator-based selection, CH-MOGP becomes more efficient for ROC convex hull approximation than algorithms that compute all Pareto optimal points. Empirical studies are conducted to compare CH-MOGP to both existing machine learning approaches and multiobjective genetic programming (MOGP) methods with classical selection schemes. Experimental results show that CH-MOGP outperforms the other approaches significantly.

Measures of novelty and interestingness are frequently encountered in the context of developmental robotics, being derived from human psychology. This work addresses these measures from the viewpoint of enhancing design-space exploration in black-box optimization. We provide a unifying notational and naming scheme with the intent of facilitating comparison, implementation, and application in the domain of design optimization. Initial analysis shows a promising interestingness measure for being tried on real-world design problems.

Evolution strategies (ESs) are powerful probabilistic search and optimization algorithms gleaned from biological evolution theory. They have been successfully applied to a wide range of real world applications. The modern ESs are mainly designed for solving continuous parameter optimization problems. Their ability to adapt the parameters of the multivariate normal distribution used for mutation during the optimization run makes them well suited for this domain. In this article we describe and study mixed integer evolution strategies (MIES), which are natural extensions of ES for mixed integer optimization problems. MIES can deal with parameter vectors consisting not only of continuous variables but also with nominal discrete and integer variables. Following the design principles of the canonical evolution strategies, they use specialized mutation operators tailored for the aforementioned mixed parameter classes. For each type of variable, the choice of mutation operators is governed by a natural metric for this variable type, maximal entropy, and symmetry considerations. All distributions used for mutation can be controlled in their shape by means of scaling parameters, allowing self-adaptation to be implemented. After introducing and motivating the conceptual design of the MIES, we study the optimality of the self-adaptation of step sizes and mutation rates on a generalized (weighted) sphere model. Moreover, we prove global convergence of the MIES on a very general class of problems. The remainder of the article is devoted to performance studies on artificial landscapes (barrier functions and mixed integer NK landscapes), and a case study in the optimization of medical image analysis systems. In addition, we show that with proper constraint handling techniques, MIES can also be applied to classical mixed integer nonlinear programming problems.

The Capacitated Vehicle Routing Problem (CVRP) is a widely studied, NP-hard problem with many real-world applications. Exact approaches are infeasible for solving large problem instances, due to the superpolynomial time complexity. Therefore, most solution approaches over the years have been metaheuristics, such as the Genetic Algorithm (GA). This paper presents a Hybrid GA, which incorporates problem-specific heuristics and domain knowledge into the algorithm. This causes the parameter settings to behave slightly different from regular GAs. To take care of the parameterization, an additional GA is used, acting on the Hybrid GA. This will be referred to as the Meta-GA. In addition to solving all known problems optimally or within 1% of the optimum, it manages to find a new best result within research literature for M-n200-k16, one of the largest problem instances.

The expected improvement (EI) is a well established criterion in Bayesian global optimization (BGO) and metamodel assisted evolutionary computation, both applied in optimization with costly function evaluations. Recently, it has been adopted in different ways to multiobjective optimization. A promising approach to formulate the expected improvement in this context, is to base it on the hypervolume indicator. Given the Bayesian model of the optimization landscape, the EI in hypervolume computes the expected gain in attained hypervolume for a given input point. Although a formulation of this expected improvement is relatively straightforward, its computation and mathematical properties are still to be investigated. This paper will outline and derive an algorithm for the exact computation of the proposed hypervolume-based EI. Moreover, this paper establishes monotonicity properties of the expected improvement. In particular the effect of the predictive distribution's variance on the hypervolume-based EI and elementary properties of the EI landscape are studied. The monotonicity properties will reveal regions where Pareto front approximations can be improved as well as underexplored regions that are favored by the hypervolume based expected improvement. A first numerical example is included that illustrates the behavior of the hypervolume-based EI in the multiobjective BGO framework.

The design of software architecture is one of the difficult tasks in the modern component-based software development which is based on the idea that develop software systems by assembling appropriate off-the-shelf components with a well-defined software architecture. Component-based software development has achieved great success and been extensively applied to a large range of application domains from realtime embedded systems to online web-based applications. In contrast to traditional approaches, it requires software architects to address a large number of non-functional requirements that can be used to quantify the operation of system. Moreover, these quality attributes can be in conflict with each other. In practice, software designers try to come up with a set of different architectural designs and then identify good architectures among them. With the increasing scale of architecture, this process becomes time-consuming and error-prone. Consequently architects could easily end up with some suboptimal designs because of large and combinatorial search space. In this paper, we introduce AQOSA (Automated Quality-driven Optimization of Software Architecture) toolkit, which integrates modeling technologies, performance analysis techniques, and advanced evolutionary multiobjective optimization algorithms (i.e. NSGA-II, SPEA2, and SMS-EMOA) to improve non-functional properties of systems in an automated manner.

Algorithms that search for robust optima often evaluate the effective fitness (robust fitness) based on stochastic approximation schemes. In this setting, finding robust optima can be recast as an optimization problem with an/a uncertain/noisy objective function. This paper studies whether state-of-the-art uncertainty handling techniques, proposed in the context of optimizing noisy objective functions, can be applied for finding robust optima. In this paper, the UH-CMA-ES is modified to handle approximations of the effective fitness. This modified approach, named RO-UH-CMA-ES, is evaluated empirically and compared to other schemes that aim to find robust optima. The experiments on multiple benchmark problems show that the RO-UH-CMA-ES yields comparable results for multi-modal problems and it outperforms other schemes on unimodal problems.

This paper presents a study for using Kriging metamodeling in combination with Covariance Matrix Adaptation Evolution Strategies (CMA-ES) to find robust solutions. A general, archive based, framework is proposed for integrating Kriging within CMA-ES, including a method to utilize the covariance matrix of the CMA-ES in a straightforward way to improve the accuracy of the Kriging predictions without introducing much additional computational cost. Moreover, it adopts an elegant way to select appropriate archive points for building a local metamodel. The study shows that this Kriging metamodeling scheme for finding robust solutions outperforms common, straightforward approaches and is very useful when there is a limited budget of function evaluations. Though using the covariance matrix can improve the prediction quality, it has no significant effect on the overall quality of the optimization results.

BACKGROUND: G protein-coupled receptors (GPCRs) represent a family of well-characterized drug targets with significant therapeutic value. Phylogenetic classifications may help to understand the characteristics of individual GPCRs and their subtypes. Previous phylogenetic classifications were all based on the sequences of receptors, adding only minor information about the ligand binding properties of the receptors. In this work, we compare a sequence-based classification of receptors to a ligand-based classification of the same group of receptors, and evaluate the potential to use sequence relatedness as a predictor for ligand interactions thus aiding the quest for ligands of orphan receptors. RESULTS: We present a classification of GPCRs that is purely based on their ligands, complementing sequence-based phylogenetic classifications of these receptors. Targets were hierarchically classified into phylogenetic trees, for both sequence space and ligand (substructure) space. The overall organization of the sequence-based tree and substructure-based tree was similar; in particular, the adenosine receptors cluster together as well as most peptide receptor subtypes (e.g. opioid, somatostatin) and adrenoceptor subtypes. In ligand space, the prostanoid and cannabinoid receptors are more distant from the other targets, whereas the tachykinin receptors, the oxytocin receptor, and serotonin receptors are closer to the other targets, which is indicative for ligand promiscuity. In 93% of the receptors studied, de-orphanization of a simulated orphan receptor using the ligands of related receptors performed better than random (AUC > 0.5) and for 35% of receptors de-orphanization performance was good (AUC > 0.7). CONCLUSIONS: We constructed a phylogenetic classification of GPCRs that is solely based on the ligands of these receptors. The similarities and differences with traditional sequence-based classifications were investigated: our ligand-based classification uncovers relationships among GPCRs that are not apparent from the sequence-based classification. This will shed light on potential cross-reactivity of GPCR ligands and will aid the design of new ligands with the desired activity profiles. In addition, we linked the ligand-based classification with a ligand-focused sequence-based classification described in literature and proved the potential of this method for de-orphanization of GPCRs.

Abstract Grid computing emerges as an infrastructure for large‐scale data processing, resource sharing, and scientific computing. In this paper we propose a Grid scheduling algorithm using multi‐attribute utility theory and multi‐objective optimization (MOO). The algorithm makes the optimal decisions based on the available set of objectives. By comparing with a deadline‐and‐budget algorithm with three objectives, we show that the proposed MOO scheduling algorithm is capable of obtaining a broader set of non‐dominated solutions. The obtained solutions are also of higher quality, which are in close proximity to the Pareto optimal front. Copyright © 2009 John Wiley & Sons, Ltd.

While the motivation and usefulness of niching methods is beyond doubt, the relaxation of assumptions and limitations concerning the hypothetical search landscape is much needed if niching is to be valid in a broader range of applications. Upon the introduction of radii-based niching methods with derandomized evolution strategies (ES), the purpose of this study is to address the so-called niche radius problem. A new concept of an adaptive individual niche radius is applied to niching with the covariance matrix adaptation evolution strategy (CMA-ES). Two approaches are considered. The first approach couples the radius to the step size mechanism, while the second approach employs the Mahalanobis distance metric with the covariance matrix mechanism for the distance calculation, for obtaining niches with more complex geometrical shapes. The proposed approaches are described in detail, and then tested on high-dimensional artificial landscapes at several levels of difficulty. They are shown to be robust and to achieve satisfying results.

There exist several applications of multi-objective evolutionary algorithms for drug design, however, a common drawback in recent approaches is that the diversity of resulting molecule populations is relatively low. This paper seeks to overcome this problem by introducing niching as a technique to enhance search space diversity. A single population approach with dynamic niche identification is studied in the application domain. In order to apply niching in molecular spaces a metric for measuring the dissimilarity of molecules will be introduced. The approach will be validated in case studies and compared with results of an NSGA-II algorithm without niching in the search space.

This paper discusses mixed integer evolution strategies (MIES) assisted by metamodels based on radial basis function networks (RBFN). The goal is to make MIES more suitable for optimization with time consuming evaluation functions.

Existing niching techniques commonly use the Euclidean distance metric in the decision space for the classification of feasible solutions to the niches under formation. This approach is likely to encounter problems in high-dimensional landscapes with non-isotropic basins of attraction. Here we consider niching with the covariance matrix adaptation evolution strategy (CMA-ES), and introduce the Mahalanobis distance metric into the niching mechanism, aiming to allow a more accurate spatial classification, based on the ellipsoids of the distribution, rather than hyper-spheres of the Euclidean metric. This is tested with the CMA-(+) routines, and compared to two niching frameworks - fixed niche radius as well as self- adaptive niche radius, which is based on the coupling to the step-size. The performance of the different variants is evaluated on a suite of theoretical test-functions. We thus present here the Mahalanobis-assisted CMA-niching as a state-of-the-art niching technique within evolution strategies (ES), and propose it as a solution to the so-called niche radius problem.

This study introduces the multi-criteria approach to the optimization of dynamic molecular alignment by shaped femtosecond laser pulses, which has been considered so far only as a single-criterion problem. The paper applies advanced Pareto front approximation algorithms to this challenging real-world, high-dimensional, and computationally expensive problem, working with low-dimensional parameterizations of the electric field. Standard approaches (NSGA-II) and their metamodel-assisted extensions based on Kriging, are applied to this optimization task and compared among each other. The study confirms the conflicting nature of the objectives. Interesting features of the problem domain, such as the geometry of the Pareto front are revealed. Furthermore, metamodel-assistance, in particular pre-screening with the Kriging-based expected improvement criterion, proves to be a valuable ingredient for improving the numerical results.

In this paper we compare Mixed-Integer Evolution Strategies (MI-ES)and standard Evolution Strategies (ES)when applied to find optimal solutions for artificial test problems and medical image processing problems. MI-ES are special instantiations of standard ES that can solve optimization problems with different objective variable types (continuous, integer, and nominal discrete). Artificial test problems are generated with a mixed-integer test generator.The practical image processing problem iss the detection of the lumen boundary in IntraVascular UltraSound (IVUS)images. Based on the experimental results, it is shown that MI-ES generally perform better than standard ES on both artifical and practical image processing problems. Moreover it is shown that MI-ES can effectively improve the parameters settings for the IVUS lumen detection algorithm.

This paper presents and analyzes in detail an efficient search method based on evolutionary algorithms (EA) assisted by local Gaussian random field metamodels (GRFM). It is created for the use in optimization problems with one (or many) computationally expensive evaluation function(s). The role of GRFM is to predict objective function values for new candidate solutions by exploiting information recorded during previous evaluations. Moreover, GRFM are able to provide estimates of the confidence of their predictions. Predictions and their confidence intervals predicted by GRFM are used by the metamodel assisted EA. It selects the promising members in each generation and carries out exact, costly evaluations only for them. The extensive use of the uncertainty information of predictions for screening the candidate solutions makes it possible to significantly reduce the computational cost of singleand multiobjective EA. This is adequately demonstrated in this paper by means of mathematical test cases and a multipoint airfoil design in aerodynamics

The S-metric or hypervolume measure is a distinguished quality measure for solution sets in Pareto optimisation. Once the aim to reach a high S-metric value is appointed, it seems to be promising to directly incorporate it in the optimisation algorithm. This idea has been implemented in the SMS-EMOA, an evolutionary multi-objective optimisation algorithm (EMOA) using the hypervolume measure within its selection operator. Solutions are rated according to their contribution to the dominated hypervolume of the current population. Up to now, the SMS-EMOA has only been applied to functions with two objectives. The work at hand extends these studies, by surveying the behaviour of the algorithm on three-objective problems. Additionally, a new efficient algorithm for the computation of the contributions to the dominated hypervolume in three-dimensional solution spaces is presented. Different variants of selection operators are proposed. Among these, a new one is presented that rates a solution concerning the number of solutions dominating it. So, solutions in less explored regions are preferred. This rating is an efficient alternative to the S-metric criterion whenever a selection among dominated solutions has to be made. Comparative studies on standard benchmark problems show that the SMS-EMOA clearly outperforms other well established EMOA. First results on a challenging real-world problem have been obtained, namely the multipoint design of an airfoil involving three objectives and nonlinear constraints. Not only a clear improvement of the baseline design but a good coverage of the Pareto front with a small limited number of points has been achieved.

The impact of operator disruption and genetic drift on the extinction of EA subpopulations on multimodal landscapes is estimated by means of idealized two-peak landscape models. To establish upper and lower bounds for extinction times the behavior of an EA that employs (μpluskommaλ) selection and recombination mechanisms is studied, assuming disruptive recombination. Markov chain and statistical simulation studies reveal that panmictic selection mechanisms as used in evolution strategies (ES) do not allow for maintaining several populations of similar fitness at the same time. Moreover, when using comma selection, good individuals might easily get lost if forming the minority of a population, an effect seemingly amplified by recombination. Niching techniques are suggested to facilitate coexistence of populations on distant attractors; conducted studies confirm their aptitude.

This paper describes the adaptation of evolutionary algorithms (EAs) to the structural optimization of chemical engineering plants, using rigorous process simulation combined with realistic costing procedures to calculate target function values. To represent chemical engineering plants, a network representation with typed vertices and variable structure will be introduced. For this representation, we introduce a technique on how to create problem specific search operators and apply them in stochastic optimization procedures. The applicability of the approach is demonstrated by a reference example. The design of the algorithms will be oriented at the systematic framework of metric-based evolutionary algorithms (MBEAs). MBEAs are a special class of evolutionary algorithms, fulfilling certain guidelines for the design of search operators, whose benefits have been proven in theory and practice. MBEAs rely upon a suitable definition of a metric on the search space. The definition of a metric for the graph representation will be one of the main issues discussed in this paper. Although this article deals with the problem domain of chemical plant optimization, the algorithmic design can be easily transferred to similar network optimization problems. A useful distance measure for variable dimensionality search spaces is suggested.