Field Of Evolutionary Multiobjective Optimization Research Articles

An Improved Local Search Method for Large-Scale Hypervolume Subset Selection

Hypervolume subset selection (HSS) has received considerable attention in the field of evolutionary multi-objective optimization (EMO). It aims to select a representative subset from a candidate solution set so that the hypervolume of the selected subset is maximized. A number of HSS methods have been proposed in the literature, attempting to either reduce the computation time of subset selection or improve the subset quality (i.e., the hypervolume of the selected subset). However, when selecting from a large candidate set (e.g., from hundreds of thousands of candidate solutions), most HSS methods fail to strike a balance between the computation time and the subset quality. In this paper, we propose a new local search HSS method and its extended version. Three strategies are proposed: The first two strategies are applied to the proposed method to obtain a good subset within a small computation time, and the third one is applied to the extended version to further improve the obtained subset. Experimental results on various candidate sets demonstrate that the proposed method and its extended version are much more efficient and effective than the existing HSS methods.

Dynamic Neighborhood Adjustment Strategy for Multi-Objective Evolutionary Algorithm Based on Decomposition

Multi-objective evolutionary algorithm based on decomposition (MOEA/D) has achieved great success in the field of evolutionary multi-objective optimization. It decomposes a multi-objective optimization problem into a number of scalar optimization sub-problems. Each sub-problem is optimized by using information from its neighboring sub-problems. Therefore, the neighborhood size of each sub-problem plays an important role in MOEA/D. Different neighborhood sizes are tested in this paper. Experimental results demonstrate that larger neighborhood size helps achieve better convergence and diversity with more CPU time and vice versa. MOEA/D uses constant neighborhood size during the whole process, and it is difficult to balance the convergence, diversity and running time. Therefore, this paper propose an algorithm based on MOEA/D. The algorithm adjusts the neighborhood size dynamically in different generations and different sub-problems to reduce the running time while the convergence and diversity of this algorithm are similar or better than other state-of-the-art algorithms. Compared to the original MOEA/D, experimental results show that adjusting the neighborhood size dynamically is a good way to reduce the running time significantly while maintaining the convergence and diversity. Furthermore, the algorithm proposed in this paper is compared with five state-of-the-art algorithms based on MOEA/D. The experimental results show that the proposed algorithm outperforms the others in efficiency while performs similarly in convergence and diversity.

MCDM, EMO and Hybrid Approaches: Tutorial and Review

Most of the practical applications that require optimization often involve multiple objectives. These objectives, when conflicting in nature, pose both optimization as well as decision-making challenges. An optimization procedure for such a multi-objective problem requires computing (computer-based search) and decision making to identify the most preferred solution. Researchers and practitioners working in various domains have integrated computing and decision-making tasks in several ways, giving rise to a variety of algorithms to handle multi-objective optimization problems. For instance, an a priori approach requires formulating (or eliciting) a decision maker’s value function and then performing a one-shot optimization of the value function, whereas an a posteriori decision-making approach requires a large number of diverse Pareto-optimal solutions to be available before a final decision is made. Alternatively, an interactive approach involves interactions with the decision maker to guide the search towards better solutions (or the most preferred solution). In our tutorial and survey paper, we first review the fundamental concepts of multi-objective optimization. Second, we discuss the classic interactive approaches from the field of Multi-Criteria Decision Making (MCDM), followed by the underlying idea and methods in the field of Evolutionary Multi-Objective Optimization (EMO). Third, we consider several promising MCDM and EMO hybrid approaches that aim to capitalize on the strengths of the two domains. We conclude with discussions on important behavioral considerations related to the use of such approaches and future work.

Benchmarking large-scale subset selection in evolutionary multi-objective optimization

In the field of evolutionary multi-objective optimization (EMO), the standard practice is to present the final population of an EMO algorithm as the output. However, it has been shown that the final population often includes solutions which are dominated by other solutions generated and discarded in previous generations. Recently, a novel EMO framework has been developed to solve this issue by storing all the non-dominated solutions generated during the evolution in an archive, and selecting a subset of solutions from the archive as the output. The key component of this framework is the subset selection from the archive, which typically stores a large number of candidate solutions. However, most relevant studies have focused on small candidate solution sets for environmental selection. There is no benchmark test suite for large-scale subset selection. This study aims to fill this research gap by proposing a benchmark test suite for large-scale subset selection, and providing a comparison between several representative subset selection algorithms using the proposed test suite. The proposed test suite together with the benchmarking studies provides a baseline for researchers to understand, use, compare, and develop large-scale subset selection algorithms in the EMO field.

A Diversity-Enhanced Subset Selection Framework for Multimodal Multiobjective Optimization

Multimodality is commonly seen in real-world multiobjective optimization problems (MOPs). In such optimization problems, namely, multimodal MOPs (MMOPs), multiple decision vectors can be projected to the same solution in the objective space (i.e., there are multiple implementations corresponding to that solution). Therefore, the diversity in the decision space is very important for the decision maker when tackling MMOPs. Subset selection methods have been widely used in the field of evolutionary multiobjective optimization for selecting well-distributed solutions (in the objective space) to be presented to the decision maker. However, since most subset selection methods do not consider the diversity of solutions in the decision space, they are not suitable for MMOPs. In this article, we aim to clearly demonstrate the usefulness of subset selection for multimodal multiobjective optimization. We propose a novel subset selection framework that can be easily integrated into existing multimodal multiobjective optimization algorithms. By selecting a prespecified number of solutions with good diversity in both the objective and decision spaces from all the examined solutions, the proposed framework significantly improves the performance of state-of-the-art multimodal multiobjective optimization algorithms on various test problems.

Fast Greedy Subset Selection From Large Candidate Solution Sets in Evolutionary Multiobjective Optimization

Subset selection plays an important role in the field of evolutionary multiobjective optimization (EMO). Especially, in an EMO algorithm with an unbounded external archive (UEA), subset selection is an essential post-processing procedure to select a prespecified number of solutions as the final result. In this article, we discuss the efficiency of greedy subset selection for the hypervolume, inverted generational distance (IGD), and IGD plus (IGD+) indicators. Greedy algorithms usually efficiently handle the subset selection. However, when a large number of solutions are given (e.g., subset selection from tens of thousands of solutions in a UEA), they often become time consuming. Our idea is to use the submodular property, which is known for the hypervolume indicator, to improve their efficiency. First, we prove that the IGD and IGD+ indicators are also submodular. Next, based on the submodular property, we propose an efficient greedy inclusion algorithm for each indicator. We demonstrate through computational experiments that the proposed algorithms are much faster than the standard greedy subset selection algorithms. The proposed algorithms also help the research on performance indicators.

A Survey on the Hypervolume Indicator in Evolutionary Multiobjective Optimization

Hypervolume is widely used as a performance indicator in the field of evolutionary multiobjective optimization (EMO). It is used not only for performance evaluation of EMO algorithms (EMOAs) but also in indicator-based EMOAs to guide the search. Since its initial proposal in the late 1990s, a wide variety of studies have been done on various topics, including hypervolume calculation, optimal μ-distribution, subset selection, hypervolume-based EMOAs, and extensions of the hypervolume indicator. However, currently there is no work to systematically survey the hypervolume indicator for these topics whereas it has been frequently used in the EMO field. This article aims to fill this gap and provide a comprehensive survey on the hypervolume indicator. We expect that this survey will help EMO researchers to understand the hypervolume indicator more deeply and thoroughly, and promote further utilization of the hypervolume indicator in the EMO field.

Reverse Strategy for Non-Dominated Archiving

In the field of evolutionary multi-objective optimization (EMO), most EMO algorithms try to find a set of non-dominated solutions to approximate the Pareto front of a multi-objective optimization problem. In these algorithms, a population is evolved from one generation to another, and the population of the last generation is presented as the final result. However, recent studies reveal that some good solutions can be discarded during the evolutionary process, whereas these solutions are non-dominated. One way to solve this issue is to store all non-dominated solutions in an unbounded external archive (UEA) during the evolutionary process and select a set of solutions from the UEA as the final result. A recently proposed ND-Tree approach is very efficient for updating the UEA whenever a new solution is generated. However, this may not be the most efficient strategy. In this paper, we propose a simple yet efficient update strategy for the ND-Tree approach. The main idea is to reverse the order of solutions with respect to their generated time when updating the UEA. The experimental results suggest that the ND-Tree approach assisted by the proposed reverse strategy is much faster than the original ND-Tree approach in obtaining the final UEA. The optimal update frequency for the proposed strategy is also investigated.

Exploring Innovative Design Principles using Innovization in Multi- Objective Optimal Reactive Power Dispatch Problem

Recently, researchers in the field of Evolutionary Multi-Objective Optimization give a systematic approach for exploring innovative design principles in a conflicting multi-objective optimization problem by analyzing pareto optimal solution using either manual or automated approach. They call it “Innovization” and defined as: “innovation through optimization”. This paper applies manual innovization to multi-objective Optimal Reactive Power Dispatch Problem using Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) as multi-objective optimization algorithm and manually searches for all possible innovative design principles by analyzing multiple pareto optimal solutions. Standard IEEE 30 bus test system is considered for the current work. Simulation results reveal number of innovative design principles. Innovative design principles includes function approximation of relationship between conflicting objectives, characteristics of decision variable with respect to different objectives and actual range of decision variables. Simulation results clearly show much faster convergence when decision variables were obtained using innovative design principles for specified desired objectives as compared to normal case. Results also show some decision variables can be eliminated by setting it to a fixed value, which leads to simplification of optimization problem as the values of these variables remains constant with respect to the values of objective function.

Adaptive region adjustment to improve the balance of convergence and diversity in MOEA/D

The multiobjective evolutionary algorithm based on decomposition (MOEA/D), which decomposes a multiobjective optimization problem (MOP) into a number of optimization subproblems and optimizes them in a collaborative manner, becomes more and more popular in the field of evolutionary multiobjective optimization. The mechanism of balance convergence and diversity is very important in MOEA/D. In the process of optimization, the chosen solutions must be distinctive and as close as possible to the Pareto front. In this paper, we first explore the relation between subproblems and solutions. Then we propose the adaptive region adjustment strategy to balance the convergence and diversity based on the objective region partition concept. Finally, this strategy is embedded in the MOEA/D framework and then a simple but efficient algorithm is proposed. To demonstrate the effectiveness of the proposed algorithm, comprehensive experiments have been designed. The simulation results show the effectiveness of our proposed algorithms.

A diversity indicator based on reference vectors for many-objective optimization

Diversity estimation of Pareto front (PF) approximations is a critical issue in the field of evolutionary multiobjective optimization. However, the existing diversity indicators are usually inappropriate for PF approximations with more than three objectives. Many of them can be utilized only when compared with approximations obtained by multiple multiobjective optimizers, which makes them difficult to use online. In this paper, we propose a unary diversity indicator based on reference vectors (DIR) to estimate the diversity of PF approximations for many-objective optimization. In DIR, a set of uniform and widespread reference vectors are generated. The coverage of each solution in the objective space is evaluated by the number of representative reference vectors it is associated with. The diversity (both spread and uniformity) is determined by the standard deviation of the coverage for all the solutions. The smaller value of DIR, the better the diversity of a PF approximation is. DIR can be applied to a unary approximation without any compared approximations needed. Thus, DIR is easy to use as either an offline indicator to estimate the performance of an optimizer or an online indicator for the selection of solutions in a MOEA. In the experimental studies, both the artificial and the real PF approximations generated by seven different many-objective algorithms are used to verify DIR as an offline indicator. The effects of the number of reference vectors on DIR are also investigated. In addition, as an online indicator, DIR is integrated into a Pareto-dominance-based evolutionary multiobjective optimizer, NSGA-II. The experimental studies show it has the significant performance enhancements over the original NSGA-II on many-objective optimization problems.

Novel performance metrics for robust multi-objective optimization algorithms

Performance metrics are essential for quantifying the performance of optimization algorithms in the field of evolutionary multi-objective optimization. Such metrics allow researchers to compare different algorithms quantitatively. In the field of robust multi-objective optimization, however, there is currently no performance metric despite its significant importance. This motivates our proposal of three novel specific metrics for measuring the convergence, coverage, and success rate of robust Pareto optimal solutions obtained by robust multi-objective algorithms. The proposed metrics are employed to quantitatively evaluate and compare Robust Multi-objective Particle Swarm Optimization (RMOPSO) and Robust Non-dominated Sorting Genetic Algorithm (RNSGA-II) on seven selected benchmark problems. The results show that the proposed metrics are effective in quantifying the performance of robust multi-objective algorithms in terms of convergence, coverage, and the ratio of the robust/non-robust Pareto optimal solutions obtained.

MOEA/D with opposition-based learning for multiobjective optimization problem

Multiobjective evolutionary algorithm based on decomposition (MOEA/D) has attracted a great deal of attention and has obtained enormous success in the field of evolutionary multiobjective optimization. It converts a multiobjective optimization problem (MOP) into a set of scalar optimization subproblems and then uses the evolutionary algorithm (EA) to optimize these subproblems simultaneously. However, there is a great deal of randomness in MOEA/D. Researchers in the field of evolutionary algorithm, reinforcement learning and neural network have reported that the simultaneous consideration of randomness and opposition has an advantage over pure randomness. A new scheme, called opposition-based learning (OBL), has been proposed in the machine learning field. In this paper, OBL has been integrated into the framework of MOEA/D to accelerate its convergence speed. Hence, our proposed approach is called opposition-based learning MOEA/D (MOEA/D-OBL). Compared with MOEA/D, MOEA/D-OBL uses an opposition-based initial population and opposition-based learning strategy to generate offspring during the evolutionary process. It is compared with its parent algorithm MOEA/D on four representative kinds of MOPs and many-objective optimization problems. Experimental results indicate that MOEA/D-OBL outperforms or performs similar to MOEA/D. Moreover, the parameter sensitivity of generalization opposite point and the probable to use OBL is experimentally investigated.

MOEA/D with Adaptive Weight Adjustment

Recently, MOEA/D (multi-objective evolutionary algorithm based on decomposition) has achieved great success in the field of evolutionary multi-objective optimization and has attracted a lot of attention. It decomposes a multi-objective optimization problem (MOP) into a set of scalar subproblems using uniformly distributed aggregation weight vectors and provides an excellent general algorithmic framework of evolutionary multi-objective optimization. Generally, the uniformity of weight vectors in MOEA/D can ensure the diversity of the Pareto optimal solutions, however, it cannot work as well when the target MOP has a complex Pareto front (PF; i.e., discontinuous PF or PF with sharp peak or low tail). To remedy this, we propose an improved MOEA/D with adaptive weight vector adjustment (MOEA/D-AWA). According to the analysis of the geometric relationship between the weight vectors and the optimal solutions under the Chebyshev decomposition scheme, a new weight vector initialization method and an adaptive weight vector adjustment strategy are introduced in MOEA/D-AWA. The weights are adjusted periodically so that the weights of subproblems can be redistributed adaptively to obtain better uniformity of solutions. Meanwhile, computing efforts devoted to subproblems with duplicate optimal solution can be saved. Moreover, an external elite population is introduced to help adding new subproblems into real sparse regions rather than pseudo sparse regions of the complex PF, that is, discontinuous regions of the PF. MOEA/D-AWA has been compared with four state of the art MOEAs, namely the original MOEA/D, Adaptive-MOEA/D, [Formula: see text]-MOEA/D, and NSGA-II on 10 widely used test problems, two newly constructed complex problems, and two many-objective problems. Experimental results indicate that MOEA/D-AWA outperforms the benchmark algorithms in terms of the IGD metric, particularly when the PF of the MOP is complex.

On Set-Based Multiobjective Optimization

Assuming that evolutionary multiobjective optimization (EMO) mainly deals with set problems, one can identify three core questions in this area of research: 1) how to formalize what type of Pareto set approximation is sought; 2) how to use this information within an algorithm to efficiently search for a good Pareto set approximation; and 3) how to compare the Pareto set approximations generated by different optimizers with respect to the formalized optimization goal. There is a vast amount of studies addressing these issues from different angles, but so far only a few studies can be found that consider all questions under one roof. This paper is an attempt to summarize recent developments in the EMO field within a unifying theory of set-based multiobjective search. It discusses how preference relations on sets can be formally defined, gives examples for selected user preferences, and proposes a general preference-independent hill climber for multiobjective optimization with theoretical convergence properties. Furthermore, it shows how to use set preference relations for statistical performance assessment and provides corresponding experimental results. The proposed methodology brings together preference articulation, algorithm design, and performance assessment under one framework and thereby opens up a new perspective on EMO.

Objective Reduction in Evolutionary Multiobjective Optimization: Theory and Applications

Many-objective problems represent a major challenge in the field of evolutionary multiobjective optimization--in terms of search efficiency, computational cost, decision making, visualization, and so on. This leads to various research questions, in particular whether certain objectives can be omitted in order to overcome or at least diminish the difficulties that arise when many, that is, more than three, objective functions are involved. This study addresses this question from different perspectives. First, we investigate how adding or omitting objectives affects the problem characteristics and propose a general notion of conflict between objective sets as a theoretical foundation for objective reduction. Second, we present both exact and heuristic algorithms to systematically reduce the number of objectives, while preserving as much as possible of the dominance structure of the underlying optimization problem. Third, we demonstrate the usefulness of the proposed objective reduction method in the context of both decision making and search for a radar waveform application as well as for well-known test functions.

Evolutionary Multiobjective Optimization in Materials Science and Engineering

This article provides a short introduction to the evolutionary multiobjective optimization field. The first part of the article discusses the most representative multiobjective evolutionary algorithms that have been developed, from a historical perspective. In the second part of the article, some representative applications within materials science and engineering are reviewed. In the final part of the article, some potential areas for future research in this area are briefly described.

Evolutionary computation and structural design: A survey of the state-of-the-art

Evolutionary computation is emerging as a new engineering computational paradigm, which may significantly change the present structural design practice. For this reason, an extensive study of evolutionary computation in the context of structural design has been conducted in the Information Technology and Engineering School at George Mason University and its results are reported here. First, a general introduction to evolutionary computation is presented and recent developments in this field are briefly described. Next, the field of evolutionary design is introduced and its relevance to structural design is explained. Further, the issue of creativity/novelty is discussed and possible ways of achieving it during a structural design process are suggested. Current research progress in building engineering systems’ representations, one of the key issues in evolutionary design, is subsequently discussed. Next, recent developments in constraint-handling methods in evolutionary optimization are reported. Further, the rapidly growing field of evolutionary multiobjective optimization is presented and briefly described. An emerging subfield of coevolutionary design is subsequently introduced and its current advancements reported. Next, a comprehensive review of the applications of evolutionary computation in structural design is provided and chronologically classified. Finally, a summary of the current research status and a discussion on the most promising paths of future research are also presented.