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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.

Advanced control system design for large wind turbines is becoming increasingly complex, and high-level optimization techniques are receiving particular attention as an instrument to fulfil this significant degree of design requirements. Multiobjective optimal (MOO) control, in particular, is today a popular methodology for achieving a control system that conciliates multiple design objectives that may typically be incompatible. Multiobjective optimization was a matter of theoretical study for a long time, particularly in the areas of game theory and operations research. Nevertheless, the discipline experienced remarkable progress and multiple advances over the last two decades. Thus, many high-complexity optimization algorithms are currently accessible to address current control problems in systems engineering. On the other hand, utilizing such methods is not straightforward and requires a long period of trying and searching for, among other aspects, start parameters, adequate objective functions, and the best optimization algorithm for the problem. Hence, the primary intention of this work is to investigate old and new MOO methods from the application perspective for the purpose of control system design, offering practical experience, some open topics, and design hints. A very challenging problem in the system engineering application of power systems is to dominate the dynamic behavior of very large wind turbines. For this reason, it is used as a numeric case study to complete the presentation of the paper.

The dynamic multiobjective evolutionary algorithm (DMOEA) is an efficient solver for dynamic multiobjective optimization problems (DMOPs). It is challenging for algorithms to converge quickly and maintain diversity in new environments. However, existing DMOEAs incorporate strategies only in the environment response stage, which may limit the further improvement of the algorithm performance. To address this problem, different strategies have been proposed in the environment response stage and static optimization stage to balance convergence and diversity throughout the optimization process. In the static optimization stage, nondominated solutions-guided evolution leaves the individuals that are closer to the nondominated individuals among the original individuals and the opposition individuals generated by opposition-based learning, which can accelerate the convergence of each generation. In the environment response stage, based on the individual dominance relationship before environmental change, the fine prediction strategy performs difference prediction and opposition-based learning prediction for nondominated and dominated individuals, respectively, which results in an initial population with good convergence and diversity in the new environment. The performance of proposed algorithm was evaluated on 22 instances and compared to eight state-of-the-art algorithms. The results show that proposed algorithm outperforms its competitors on most problems. Additionally, proposed algorithm performs better in early environmental changes and is relatively insensitive to different severity changes.

In this paper, we investigate a robust nonsmooth multiobjective optimization problem related to a multiobjective optimization with data uncertainty. We firstly introduce two kinds of generalized convex functions, which are not necessary to be convex. Robust necessary optimality conditions for weakly robust efficient solutions and properly robust efficient solutions of the problem are established by a generalized alternative theorem and the robust constraint qualification. Further, robust sufficient optimality conditions for weakly robust efficient solutions and properly robust efficient solutions of the problem are also derived. The Mond–Weir-type dual problem and Wolfe-type dual problem are formulated. Finally, we obtain the weak, strong and converse robust duality results between the primal one and its dual problems under the generalized convexity assumptions.

Previously, cluster-based multi or many objective function techniques were proposed to reduce the Pareto set. Recently, researchers proposed such techniques to find better solutions in the objective space to solve engineering problems. In this work, we applied a cluster-based approach for solution selection in a multiobjective evolutionary algorithm based on decomposition with bare bones particle swarm optimization for data clustering and investigated its clustering performance. In our previous work, we found that MOEA/D with BBPSO performed the best on 10 datasets. Here, we extend this work applying a cluster-based approach tested on 13 UCI datasets. We compared with six multiobjective evolutionary clustering algorithms from the existing literature and ten from our previous work. The proposed technique was found to perform well on datasets highly overlapping clusters, such as CMC and Sonar. So far, we found only one work that used cluster-based MOEA for clustering data, the hierarchical topology multiobjective clustering algorithm. All other cluster-based MOEA found were used to solve other problems that are not data clustering problems. By clustering Pareto solutions and evaluating new candidates against the found cluster representatives, local search is introduced in the solution selection process within the objective space, which can be effective on datasets with highly overlapping clusters. This is an added layer of search control in the objective space. The results are found to be promising, prompting different areas of future research which are discussed, including the study of its effects with an increasing number of clusters as well as with other objective functions.

In this paper, we obtain approximate necessary and sufficient optimality conditions, characterizing an approximately efficient solution of a semi-infinite multiobjective fractional problem under the closedness qualification condition. As a consequence, we derive approximate necessary and sufficient optimality conditions characterizing an approximately efficient solution for a constrained multiobjective fractional programming problem. Furthermore, we present examples illustrating our main results.

The expected improvement (EI) algorithm is a very popular method for expensive optimization problems. In the past twenty years, the EI criterion has been extended to deal with a wide range of expensive optimization problems. This paper gives a comprehensive review of the EI extensions designed for parallel optimization, multiobjective optimization, constrained optimization, noisy optimization, multi-fidelity optimization and high-dimensional optimization. The main challenges of extending the EI approach to solve these complex optimization problems are pointed out, and the ideas proposed in literature to tackle these challenges are highlighted. For each reviewed algorithm, the surrogate modeling method, the computation of the infill criterion and the internal optimization of the infill criterion are carefully studied and compared. In addition, the monotonicity properties of the multiobjective EI criteria and constrained EI criteria are analyzed in detail. Through this review, we give an organized summary about the EI developments in the past twenty years and show a clear picture about how the EI approach has advanced. In the end of this paper, several interesting problems and future research topics about the EI developments are given.

In the real world, multiobjective optimization problems require the efficient acquisition of diverse solutions. Various multiobjective evolutionary algorithms (MOEAs) have been developed to address these problems. Typically, MOEAs use the same scoring criteria for both survival and mating selection, despite their different roles. Survival selection should ensure convergence and diversity, whereas mating selection should focus on selecting individuals with higher convergence for crossover. In this article, an efficient selection algorithm is proposed that integrates data envelopment analysis (DEA), Pareto front modeling, and a reference crossover mechanism. In survival selection, algorithms are used to ensure high convergence and diversity. In a previous study, DEA was employed to select individuals with higher convergence in mating selection. This approach balances convergence and diversity. In addition, Pareto front modeling addresses the convexity assumption issue in DEA. In this study, by selecting constraint solutions obtained through DEA as crossover targets, the algorithm makes crossover with superior solutions possible, enhancing optimization speed and diversity. The algorithm is particularly effective for benchmark functions that benefit from neighborhood crossover. In comparisons using the hypervolume metric on the WFG and DTLZ benchmark functions, the proposed algorithm outperformed NSGA-II, NSGA-III, AGE-MOEA-II, DEA-GA, MOEA/D, and other previous algorithms. The results of a Wilcoxon rank-sum test also showed that the proposed algorithm is statistically superior.

Multiobjective reliability-based design optimization (RBDO) is a research area, which has not been investigated in the literatures comparing with single-objective RBDO. This work conducts an exhaustive study of fifteen new and popular metaheuristic multiobjective RBDO algorithms, including non-dominated sorting genetic algorithm II, differential evolution for multiobjective optimization, multiobjective evolutionary algorithm based on decomposition, multiobjective particle swarm optimization, multiobjective flower pollination algorithm, multiobjective bat algorithm, multiobjective gray wolf optimizer, multiobjective multiverse optimization, multiobjective water cycle optimizer, success history-based adaptive multiobjective differential evolution, success history-based adaptive multiobjective differential evolution with whale optimization, multiobjective salp swarm algorithm, real-code population-based incremental learning and differential evolution, unrestricted population size evolutionary multiobjective optimization algorithm, and multiobjective jellyfish search optimizer. In addition, the adaptive chaos control method is employed for the above-mentioned algorithms to estimate the probabilistic constraints effectively. This comparative analysis reveals the critical technologies and enormous challenges in the RBDO field. It also offers new insight into simultaneously dealing with the multiple conflicting design objectives and probabilistic constraints. Also, this study presents the advantage and future development trends or incurs the increased challenge of researchers to put forward an effective multiobjective RBDO algorithm that assists the complex engineering system design.

The search region in multiobjective optimization problems is a part of the objective space where nondominated points could lie. It plays an important role in the generation of the nondominated set of multiobjective combinatorial optimization (MOCO) problems. In this paper, we establish the representation of the search region by half-open polyblocks (a variant concept of “polyblock” in monotonic optimization) and propose a new procedure for updating the search region. We also study the impact of stack policies to the new procedure and the existing methods that update the search region. Stack policies are then analyzed, pointing out their performance effectiveness by means of the results of rich computational experiments on finding the whole set of nondominated points of MOCO problems.