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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2019, Vol. 36 ›› Issue (1): 59-70.doi: 10.3969/j.issn.1005-3085.2019.01.005

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锥-逼近多值函数和集值优化的近似解

孔翔宇1,   余国林1,   刘三阳2   

  1. 1- 北方民族大学应用数学研究所，银川  750021
    2- 西安电子科技大学数学与统计学院，西安  710071
  • 收稿日期:2016-12-07 接受日期:2017-09-28 出版日期:2019-02-15 发布日期:2019-04-15
  • 基金资助:
    国家自然科学基金(11361001)；宁夏自然科学基金(NZ17114)；北方民族大学科研项目(2017SXKY06).

Approximating Multifunctions and Approximate Solutions in Set-valued Optimization

KONG Xiang-yu1,   YU Guo-lin1,   LIU San-yang2   

  1. 1- Institute of Applied Mathematics, Beifang University of Nationalities, Yinchuan 750021
    2- School of Mathematics and Statistics, Xidian University, Xi'an 710071
  • Received:2016-12-07 Accepted:2017-09-28 Online:2019-02-15 Published:2019-04-15
  • Supported by:
    The National Natural Science Foundation of China (11361001); the Natural Science Foundation of Ningixa (NZ17114); the 2017 Scientific Research Project of North Minzu University (2017SXKY06).

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摘要： 最优性条件和对偶理论是集值向量优化研究领域的重点问题之一．本文的目的是建立一类广义凸集值优化的最优性条件和对偶定理，在锥-逼近多值函数概念的基础上，定义集值映射的一类新的广义不变凸性，称之为次不变凸集值映射，在这类广义凸性假设下，研究最优性条件和对偶定理．利用分析的方法，本文得到了集值优化问题关于弱近似极小元的一个最优性充分条件，以及Mond-Weir和Wolfe两种模型下的弱对偶定理、强对偶定理和逆对偶定理．本文所得结果丰富和深化了集值优化理论及其应用的研究内容．

关键词: 集值优化, 不变凸性, 最优性条件, 锥-逼近多值函数, 对偶

Abstract: Optimality conditions and duality are of great importance in vector optimization with set-valued mappings. The aim of this paper is to establish the sufficient optimality condition and duality theorems for a kind of generalized convex set-valued optimization problems. Based upon the concept of invexity in terms of cone-approximating multifunction for a set-valued map, a new kind of generalized invexities, termed subinvex set-valued mappings, is introduced, and optimality conditions and duality theorems are investigated for its constraint set-valued optimization. It also presents an example to illustrate their existence. By employing the analytic method, a sufficient optimality condition and weak, strong, converse duality theorems between Mond-Weir and Wolfe dual problems and the primal constraint set-valued optimization problems are proposed in sense of weakly approximate minimizers. The results  obtained in this note enrich and deepen the theory and applications of set-valued optimization.

Key words: set-valued optimization, invexity, optimality conditions, cone-approximating multifunction, duality

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