约束优化问题中多目标进化算法的反对称修复操作器

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"这篇论文提出了一种用于多目标约束优化问题的反对称修复操作符，旨在解决违反盒约束的解决方案。修复操作符采用反向校正策略，有效地防止种群陷入局部最优。该操作符被集成到两个经典的多目标进化算法MOEA/D和NSGA-II中，以增强它们在处理约束条件下的优化能力。" 本文主要讨论的是在多目标优化问题（MCOPs）中的约束优化策略。多目标优化问题涉及寻找一组解，这些解在多个相互冲突的目标函数之间达到平衡，同时还要满足一系列约束条件。传统的优化方法在处理这类问题时可能会遇到困难，特别是在约束条件下，可能导致生成的解决方案不合法或性能不佳。论文提出了一个创新的修复操作符，其核心是反向校正策略。当解决方案违反了约束（即超出规定的盒约束）时，这种修复操作符会介入。盒约束通常是指限制决策变量的取值范围，确保解决方案在可行域内。反向校正策略的目的是通过一种相反的调整方式来修复超出约束的解，这种方法能够帮助算法避免陷入局部最优，从而提高全局探索能力。局部最优是指算法在优化过程中过早收敛到一个不是全局最优的解决方案。为了验证新修复操作符的有效性，作者将其集成到两种著名的多目标进化算法中：分解式多目标进化算法（MOEA/D）和非支配排序遗传算法II（NSGA-II）。这两种算法在多目标优化领域有着广泛的应用，但它们自身在处理约束问题时可能不够理想。通过结合修复操作符，算法能够在保持原有的优点的同时，增强处理约束的能力，生成更合规且分布均匀的帕累托前沿。此外，论文在2015年国际自然计算会议（ICNC）上发表，表明了研究的学术性和专业性。参与研究的学者来自华南理工大学、汕头大学等知名高校，以及美国密歇根州立大学的BEACON研究中心，展示了跨学科的协作与深度研究。这篇论文提出的反对称修复操作符为多目标优化问题提供了一种新的解决方案，尤其是在处理约束条件时，提高了优化算法的性能和全局搜索能力。这一方法对于优化领域的理论研究和实际应用都具有重要的参考价值。

2015 11th International Conference on Natural Computation (ICNC)

An Opposition-based Repair Operator for

Multi-objective Evolutionary Algorithm in

Constrained Optimization Problems

Zhun Fan

Department of Electronic Engineering

Shantou University

Guangdong, Shantou 515063

Han Huang

School of Software Engineering

South China University of

Technology, Guangdong, Guangzhou 510006

Wenji Li

Department of Electronic Engineering

Shantou University

Guangdong, Shantou 515063

Shuxiang Xie

Department of Electronic Engineering

Shantou University

Guangdong, Shantou 515063

Xinye Cai

College of Computer Science and

Technology, Nanjing University of

Aeronautics and Astronautics

Jiangsu, Nanjing 210016

Erik Goodman

BEACON Centre for the Study of

Evolution in Action, Michigan State

University, East Lansing, MI, 48824

Abstract—In this paper, we design a set of multi-objective

constrained optimization problems (MCOPs) and propose a new

repair operator to address them. The proposed repair operator

is used to ﬁx the solutions that violate the box constraints. More

speciﬁcally, it employs a reversed correction strategy that can

effectively avoid the population falling into local optimum. In

addition, we integrate the proposed repair operator into two

classical multi-objective evolutionary algorithms MOEA/D and

NSGA-II. The proposed repair operator is compared with other

two kinds of commonly used repair operators on benchmark

problems CTPs and MCOPs. The experiment results demon-

strate that our proposed approach is very effective in terms of

convergence and diversity.

Index Terms—Multi-objective Evolutionary Algorithm, Repair

Operator, Constrained Optimization.

I. INTRODUCTION

Multi-objective optimization problems (MOPs) consist of

more than one objectives, which are usually conﬂicting with

each other. In other words, improvements in one objective may

lead to the degradation of other objectives. It is impossible to

make all of the objectives to be optimal at the same time.

Instead, a set of solutions that represent the trade-off among

multiple objectives exist for MOPs. In addition, different types

of constraints are often unavoidable in MOPs. Such MOPs

with constraints are usually termed multi-objective constrained

optimization problem. Constraints can be roughly divided into

two categories, equality and inequality constraints. Without

loss of generality, a multi-objective constrained optimization

problem can be deﬁned as follows:

minimize F (x) = (f

(x), . . . , f

(x)) (1)

subject to g

(x) ≥ 0, i = 1, . . . , q

subject to h

(x) = 0, j = 1, . . . , p

Where x = (x

, x

, . . . , x

) ⊂ R

is n-dimensional design

variables,F(x) = (f

(x), f

(x), . . . , f

(x)) ⊂ R

is m-

dimensional objective vector. g

(x) ≥ 0 deﬁne q inequality

constraints, h

(x) = 0 deﬁne p equality constraints.

The existing multi-objective constrained evolutionary algo-

rithms combine the multi-objective evolutionary algorithms

with the mechanisms of constraint handling [1]. At present,

NSGA-II [2] and MOEA/D [3] are the two classical multi-

objective evolutionary algorithms representing two categories

of ﬁtness assignment methods, namely ﬁtness assignment

based on domination and decomposition. In ﬁtness assignment

based on domination, the ﬁtness is decided by non-dominated

sorting and crowding distance. Representative algorithms us-

ing this type of ﬁtness assignment method include MOGA

[4] , PAES-II [5] , SPEA-II [6] and NSGA-II [2]. In ﬁtness

assignment based on decomposition, comparison and sorting

of individuals are made via aggregation function with weights

allocated speciﬁcally to all individuals. Typical algorithms of

this category include IMMOGLS [7], UGA [8] , cMOGA [9]

, MOGLS [10] , and MOEA/D [3].

The existing constraints handling mechanism can be divided

into four categories. They are feasibility maintenance, penalty

function, separation of constraint violation and objective value,

and multi-objective evolutionary algorithms (MOEAs). The

methods of feasibility maintenance are usually applied to the

discrete optimization problems, such as the job shop schedul-

ing problems and the vehicle routing problems. They either

design appropriate coding and decoding methods to ensure

that the individuals are feasible, or apply some mechanisms

to repair the infeasible individuals. Unlike the feasibility main-

tenance methods, the penalty function method is adding one

penalty term to the objective functions and transforming the

constrained optimization problem into an unconstrained one.

Typical methods of this category include segregated penalty

functions [17], death penalty functions [18], co-evolutionary

penalty functions [19] and adaptive penalty functions [20,21].

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