基于Benson标量化方法的多目标优化问题解集刻画  

Characterization of Solution Set of Multiobjective Optimization Problem Based on Benson’s Method

作　　者：钟游 高英 ZHONG You;GAO Ying(School of Mathematical Sciences,Chongqing Normal University,Chongqing 401331,China)

机构地区：[1]重庆师范大学数学科学学院,重庆401331

出　　处：《重庆师范大学学报（自然科学版）》2023年第1期88-94,共7页Journal of Chongqing Normal University:Natural Science

基　　金：国家自然科学基金(No.12171063);重庆市科学技术研究重点项目(No.KJZDK202001104);重庆市高校创新研究群体项目(No.CXQT20014);重庆市自然科学基金(No.cstc2022ycjh-bgzxm0114);重庆市留学人员回国创业创新支持计划(No.cx2020096)。

摘　　要：[目的]基于Benson标量化方法研究多目标优化问题有效解集和真有效解集空性的刻画。[方法]利用标量化方法和稠密性结果研究多目标优化问题有效解集和真有效解集的空性刻画。[结果]首先得出了自然锥序下Benson标量化问题无界的等价刻画,并在此基础上给出了多目标优化问题有效解集和真有效解集为空集的必要条件。其次得到了字典序下有效解集和Borwein真有效解集为空集的条件,同时对假设条件进行举例说明。最后给出了一般锥序下Benson标量化问题无界的必要条件,以及多目标优化问题有效解和Benson标量化问题最优解的关系。[结论]针对凸和非凸多目标优化问题给出解集的空性刻画。[Purposes]To study the characterization of the emptiness of efficient solution set and properly efficient solution set for multiobjective optimization problems based on Benson’s method.[Methods]By using the scalarization method and the density results to study the characterization of the emptiness of the efficient solution set and the properly efficient solution set of multiobjective optimization problems.[Findings]Firstly,obtaining the equivalent characterization of unbounded Benson scalarization problem under natural cone order,and on this basis,giving the necessary conditions that the efficient solution set and the properly efficient solution set of multiobjective optimization problem are empty sets.Secondly,obtaining the conditions that the efficient solution set and the Borwein properly efficient solution set under dictionary order are empty sets,and give examples to illustrate the assumptions.Finally,the necessary conditions for the unbounded Benson scalar problem under the general cone order are given,and the relationship between the effective solution of the multiobjective optimization problem and the optimal solution of the Benson scalar problem is also given.[Conclusions]For convex and nonconvex multiobjective optimization problems,the emptiness of solution sets is characterized.

关 键 词：多目标优化 Benson标量化方法 解集刻画 

分 类 号：O221.2[理学—运筹学与控制论]