scholarly journals Improved Lebesgue Indicator-Based Evolutionary Algorithm: Reducing Hypervolume Computations

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 19
Author(s):  
Saúl Zapotecas-Martínez ◽  
Abel García-Nájera ◽  
Adriana Menchaca-Méndez
Keyword(s):  
Evolutionary Algorithms ◽  
Evolutionary Algorithm ◽  
Lebesgue Measure ◽  
Pareto Front ◽  
Optimization Problems ◽  
State Of The Art ◽  
Computational Cost ◽  
Computational Time ◽  
Multi Objective Optimization ◽  
Multi Objective

One of the major limitations of evolutionary algorithms based on the Lebesgue measure for multi-objective optimization is the computational cost required to approximate the Pareto front of a problem. Nonetheless, the Pareto compliance property of the Lebesgue measure makes it one of the most investigated indicators in the designing of indicator-based evolutionary algorithms (IBEAs). The main deficiency of IBEAs that use the Lebesgue measure is their computational cost which increases with the number of objectives of the problem. On this matter, the investigation presented in this paper introduces an evolutionary algorithm based on the Lebesgue measure to deal with box-constrained continuous multi-objective optimization problems. The proposed algorithm implicitly uses the regularity property of continuous multi-objective optimization problems that has suggested effectiveness when solving continuous problems with rough Pareto sets. On the other hand, the survival selection mechanism considers the local property of the Lebesgue measure, thus reducing the computational time in our algorithmic approach. The emerging indicator-based evolutionary algorithm is examined and compared versus three state-of-the-art multi-objective evolutionary algorithms based on the Lebesgue measure. In addition, we validate its performance on a set of artificial test problems with various characteristics, including multimodality, separability, and various Pareto front forms, incorporating concavity, convexity, and discontinuity. For a more exhaustive study, the proposed algorithm is evaluated in three real-world applications having four, five, and seven objective functions whose properties are unknown. We show the high competitiveness of our proposed approach, which, in many cases, improved the state-of-the-art indicator-based evolutionary algorithms on the multi-objective problems adopted in our investigation.

scholarly journals The dilemma between eliminating dominance-resistant solutions and preserving boundary solutions of extremely convex Pareto fronts

2021 ◽  
Author(s):  
Zhenkun Wang ◽  
Qingyan Li ◽  
Qite Yang ◽  
Hisao Ishibuchi
Keyword(s):  
Evolutionary Algorithms ◽  
Coping Strategies ◽  
Test Problem ◽  
Pareto Front ◽  
Optimization Problems ◽  
Feasible Region ◽  
Multi Objective Optimization ◽  
Multi Objective ◽  
Pareto Fronts ◽  
Boundary Solutions

AbstractIt has been acknowledged that dominance-resistant solutions (DRSs) extensively exist in the feasible region of multi-objective optimization problems. Recent studies show that DRSs can cause serious performance degradation of many multi-objective evolutionary algorithms (MOEAs). Thereafter, various strategies (e.g., the $$\epsilon $$ ϵ -dominance and the modified objective calculation) to eliminate DRSs have been proposed. However, these strategies may in turn cause algorithm inefficiency in other aspects. We argue that these coping strategies prevent the algorithm from obtaining some boundary solutions of an extremely convex Pareto front (ECPF). That is, there is a dilemma between eliminating DRSs and preserving boundary solutions of the ECPF. To illustrate such a dilemma, we propose a new multi-objective optimization test problem with the ECPF as well as DRSs. Using this test problem, we investigate the performance of six representative MOEAs in terms of boundary solutions preservation and DRS elimination. The results reveal that it is quite challenging to distinguish between DRSs and boundary solutions of the ECPF.

A many-objective evolutionary algorithm based on vector angle distance scaling

2021 ◽  
Vol 40 (5) ◽  
pp. 10285-10306
Author(s):  
Xin Li ◽  
Xiaoli Li ◽  
Kang Wang
Keyword(s):  
Evolutionary Algorithms ◽  
Evolutionary Algorithm ◽  
Pareto Front ◽  
Optimization Problems ◽  
Adaptive Strategy ◽  
Great Success ◽  
Multi Objective ◽  
Environmental Selection ◽  
Fitness Value ◽  
Vector Angle

In the past two decades, multi-objective evolutionary algorithms (MOEAs) have achieved great success in solving two or three multi-objective optimization problems. As pointed out in some recent studies, however, MOEAs face many difficulties when dealing with many-objective optimization problems(MaOPs) on account of the loss of the selection pressure of the non-dominant candidate solutions toward the Pareto front and the ineffective design of the diversity maintenance mechanism. This paper proposes a many-objective evolutionary algorithm based on vector guidance. In this algorithm, the value of vector angle distance scaling(VADS) is applied to balance convergence and diversity in environmental selection. In addition, tournament selection based on the aggregate fitness value of VADS is applied to generate a high quality offspring population. Besides, we adopt an adaptive strategy to adjust the reference vector dynamically according to the scales of the objective functions. Finally, the performance of the proposed algorithm is compared with five state-of-the-art many-objective evolutionary algorithms on 52 instances of 13 MaOPs with diverse characteristics. Experimental results show that the proposed algorithm performs competitively when dealing many-objective with different types of Pareto front.

scholarly journals An Ensemble Framework of Evolutionary Algorithm for Constrained Multi-Objective Optimization

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 116
Author(s):  
Junhua Ku ◽  
Fei Ming ◽  
Wenyin Gong
Keyword(s):  
Evolutionary Algorithms ◽  
Real World ◽  
Optimization Problems ◽  
State Of The Art ◽  
Multi Objective Optimization ◽  
The Real ◽  
Multi Objective ◽  
The Past ◽  
Hypervolume Indicator ◽  
Real World Problems

In the real-world, symmetry or asymmetry widely exists in various problems. Some of them can be formulated as constrained multi-objective optimization problems (CMOPs). During the past few years, handling CMOPs by evolutionary algorithms has become more popular. Lots of constrained multi-objective optimization evolutionary algorithms (CMOEAs) have been proposed. Whereas different CMOEAs may be more suitable for different CMOPs, it is difficult to choose the best one for a CMOP at hand. In this paper, we propose an ensemble framework of CMOEAs that aims to achieve better versatility on handling diverse CMOPs. In the proposed framework, the hypervolume indicator is used to evaluate the performance of CMOEAs, and a decreasing mechanism is devised to delete the poorly performed CMOEAs and to gradually determine the most suitable CMOEA. A new CMOEA, namely ECMOEA, is developed based on the framework and three state-of-the-art CMOEAs. Experimental results on five benchmarks with totally 52 instances demonstrate the effectiveness of our approach. In addition, the superiority of ECMOEA is verified through comparisons to seven state-of-the-art CMOEAs. Moreover, the effectiveness of ECMOEA on the real-world problems is also evaluated for eight instances.

scholarly journals An Entropy-Based Multiobjective Evolutionary Algorithm with an Enhanced Elite Mechanism

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Yufang Qin ◽  
Junzhong Ji ◽  
Chunnian Liu
Keyword(s):  
Evolutionary Algorithm ◽  
Pareto Front ◽  
Optimization Problem ◽  
State Of The Art ◽  
Early Stage ◽  
Test Problems ◽  
Multiobjective Optimization Problem ◽  
Multi Objective Optimization ◽  
Nsga Ii ◽  
Multi Objective

Multiobjective optimization problem (MOP) is an important and challenging topic in the fields of industrial design and scientific research. Multi-objective evolutionary algorithm (MOEA) has proved to be one of the most efficient algorithms solving the multi-objective optimization. In this paper, we propose an entropy-based multi-objective evolutionary algorithm with an enhanced elite mechanism (E-MOEA), which improves the convergence and diversity of solution set in MOPs effectively. In this algorithm, an enhanced elite mechanism is applied to guide the direction of the evolution of the population. Specifically, it accelerates the population to approach the true Pareto front at the early stage of the evolution process. A strategy based on entropy is used to maintain the diversity of population when the population is near to the Pareto front. The proposed algorithm is executed on widely used test problems, and the simulated results show that the algorithm has better or comparative performances in convergence and diversity of solutions compared with two state-of-the-art evolutionary algorithms: NSGA-II, SPEA2 and the MOSADE.

An Efficient Framework Using Normalized Dominance Operator for Multi-Objective Evolutionary Algorithms

2019 ◽  
Vol 10 (1) ◽  
pp. 15-37 ◽  
Author(s):  
Muneendra Ojha ◽  
Krishna Pratap Singh ◽  
Pavan Chakraborty ◽  
Shekhar Verma
Keyword(s):  
Evolutionary Algorithms ◽  
Pareto Front ◽  
Optimization Problem ◽  
Computational Cost ◽  
Optimization Methods ◽  
Objective Test ◽  
Test Problems ◽  
Multi Objective Optimization ◽  
Pareto Dominance ◽  
Multi Objective

Multi-objective optimization algorithms using evolutionary optimization methods have shown strength in solving various problems using several techniques for producing uniformly distributed set of solutions. In this article, a framework is presented to solve the multi-objective optimization problem which implements a novel normalized dominance operator (ND) with the Pareto dominance concept. The proposed method has a lesser computational cost as compared to crowding-distance-based algorithms and better convergence. A parallel external elitist archive is used which enhances spread of solutions across the Pareto front. The proposed algorithm is applied to a number of benchmark multi-objective test problems with up to 10 objectives and compared with widely accepted aggregation-based techniques. Experiments produce a consistently good performance when applied to different recombination operators. Results have further been compared with other established methods to prove effective convergence and scalability.

A Decomposition-based Evolutionary Algorithm with Correlative Selection Mechanism for Many-objective Optimization

2020 ◽  
pp. 1-36
Author(s):  
Ruochen Liu ◽  
Ruinan Wang ◽  
Renyu Bian ◽  
Jing Liu ◽  
Licheng Jiao
Keyword(s):  
Evolutionary Algorithms ◽  
Evolutionary Algorithm ◽  
Reference Point ◽  
Optimization Problems ◽  
State Of The Art ◽  
Reference Points ◽  
Test Problems ◽  
Selection Mechanism ◽  
Multi Objective Optimization ◽  
Decomposition Approach

Decomposition-based evolutionary algorithms have been quite successful in dealing with multi-objective optimization problems. Recently, more and more researchers attempt to apply the decomposition approach to solve many-objective optimization problems. A many-objective evolutionary algorithm based on decomposition with correlative selection mechanism (MOEA/D-CSM) is also proposed to solve many-objective optimization problems in this paper. Since MOEA/D-SCM is based on a decomposition approach which adopts penalty boundary intersection (PBI), a set of reference points must be generated in advance. Thus, a new concept related to the set of reference points is introduced first, namely, the correlation between an individual and a reference point. Thereafter, a new selection mechanism based on the correlation is designed and called correlative selection mechanism. The correlative selection mechanism finds their correlative individuals for each reference point as soon as possible so that the diversity among population members is maintained. However, when a reference point has two or more correlative individuals, those worse correlative individuals may be removed from a population so that the solutions can be ensured to move towards the Pareto-optimal front. In a comprehensive experimental study, we apply MOEA/D-CSM to a number of many-objective test problems with 3 to 15 objec-tives and make a comparison with three state-of-the-art many-objective evolutionary algorithms, namely, NSGA-III, MOEA/D and RVEA. Experimental results show that the proposed MOEA/D-CSM can produce competitive results on most of the problems considered in this study.

scholarly journals Memory-Enhanced Dynamic Multi-Objective Evolutionary Algorithm Based on Lp Decomposition

2018 ◽  
Vol 8 (9) ◽  
pp. 1673 ◽  
Author(s):  
Xinxin Xu ◽  
Yanyan Tan ◽  
Wei Zheng ◽  
Shengtao Li
Keyword(s):  
Evolutionary Algorithms ◽  
Evolutionary Algorithm ◽  
Dynamic Optimization ◽  
Decomposition Method ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Multi Objective Optimization ◽  
Scalar Optimization ◽  
Multi Objective ◽  
Memory Scheme

Decomposition-based multi-objective evolutionary algorithms provide a good framework for static multi-objective optimization. Nevertheless, there are few studies on their use in dynamic optimization. To solve dynamic multi-objective optimization problems, this paper integrates the framework into dynamic multi-objective optimization and proposes a memory-enhanced dynamic multi-objective evolutionary algorithm based on L p decomposition (denoted by dMOEA/D- L p ). Specifically, dMOEA/D- L p decomposes a dynamic multi-objective optimization problem into a number of dynamic scalar optimization subproblems and coevolves them simultaneously, where the L p decomposition method is adopted for decomposition. Meanwhile, a subproblem-based bunchy memory scheme that stores good solutions from old environments and reuses them as necessary is designed to respond to environmental change. Experimental results verify the effectiveness of the L p decomposition method in dynamic multi-objective optimization. Moreover, the proposed dMOEA/D- L p achieves better performance than other popular memory-enhanced dynamic multi-objective optimization algorithms.

Genetic Diversity as an Objective in Multi-Objective Evolutionary Algorithms

2003 ◽  
Vol 11 (2) ◽  
pp. 151-167 ◽  
Author(s):  
Andrea Toffolo ◽  
Ernesto Benini
Keyword(s):  
Genetic Diversity ◽  
Evolutionary Algorithms ◽  
Evolutionary Algorithm ◽  
Selection Pressure ◽  
Evaluation Method ◽  
State Of The Art ◽  
Search Space ◽  
Test Problems ◽  
Multi Objective ◽  
Diversity Evaluation

A key feature of an efficient and reliable multi-objective evolutionary algorithm is the ability to maintain genetic diversity within a population of solutions. In this paper, we present a new diversity-preserving mechanism, the Genetic Diversity Evaluation Method (GeDEM), which considers a distance-based measure of genetic diversity as a real objective in fitness assignment. This provides a dual selection pressure towards the exploitation of current non-dominated solutions and the exploration of the search space. We also introduce a new multi-objective evolutionary algorithm, the Genetic Diversity Evolutionary Algorithm (GDEA), strictly designed around GeDEM and then we compare it with other state-of-the-art algorithms on a well-established suite of test problems. Experimental results clearly indicate that the performance of GDEA is top-level.

Toward the Use of Pareto Performance Solutions and Pareto Robustness Solutions for Multi-Objective Robust Optimization Problems

2012 ◽  
Author(s):  
Weijun Wang ◽  
Stéphane Caro ◽  
Fouad Bennis ◽  
Oscar Brito Augusto
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Multi Objective Optimization ◽  
Robust Solutions ◽  
Multi Objective ◽  
Robust Optimization Problem ◽  
Design Solutions ◽  
Further Development

For Multi-Objective Robust Optimization Problem (MOROP), it is important to obtain design solutions that are both optimal and robust. To find these solutions, usually, the designer need to set a threshold of the variation of Performance Functions (PFs) before optimization, or add the effects of uncertainties on the original PFs to generate a new Pareto robust front. In this paper, we divide a MOROP into two Multi-Objective Optimization Problems (MOOPs). One is the original MOOP, another one is that we take the Robustness Functions (RFs), robust counterparts of the original PFs, as optimization objectives. After solving these two MOOPs separately, two sets of solutions come out, namely the Pareto Performance Solutions (PP) and the Pareto Robustness Solutions (PR). Make a further development on these two sets, we can get two types of solutions, namely the Pareto Robustness Solutions among the Pareto Performance Solutions (PR(PP)), and the Pareto Performance Solutions among the Pareto Robustness Solutions (PP(PR)). Further more, the intersection of PR(PP) and PP(PR) can represent the intersection of PR and PP well. Then the designer can choose good solutions by comparing the results of PR(PP) and PP(PR). Thanks to this method, we can find out the optimal and robust solutions without setting the threshold of the variation of PFs nor losing the initial Pareto front. Finally, an illustrative example highlights the contributions of the paper.

Sign in / Sign up

Export Citation Format

Share Document