区间值优化问题的 KKT 和弱互补近似 KKT 条件

数学物理学报 ›› 2023, Vol. 43 ›› Issue (6): 1897-1913.

区间值优化问题的 KKT 和弱互补近似 KKT 条件

黄晓美¹(),唐国吉^1,^2,^*()

¹广西民族大学数学与物理学院南宁 530006
²广西民族大学 & 广西应用数学中心 & 广西混杂计算与集成电路设计分析重点实验室南宁 530006

收稿日期:2022-12-23 修回日期:2023-03-23 出版日期:2023-12-26 发布日期:2023-11-16
通讯作者: *唐国吉，E-mail: guojvtang@126.com
作者简介:黄晓美，E-mail: hxiaomei2021@163.com
基金资助:
国家自然科学基金(11961006);广西自然科学基金(2020GXNSFAA159100)

KKT and Weakly Complementary Approximate KKT Conditions for Interval-Valued Optimization Problems

Huang Xiaomei¹(),Tang Guoji^1,^2,^*()

¹School of Mathematics and Physics, Guangxi Minzu University, Nanning 530006
²School of Mathematics and Physics & Center for Applied Mathematics of Guangxi & Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi Minzu University, Nanning 530006

Received:2022-12-23 Revised:2023-03-23 Online:2023-12-26 Published:2023-11-16
Supported by:
NSFC(11961006);NSF of Guangxi Province(2020GXNSFAA159100)

摘要/Abstract

摘要：

研究含等式和不等式约束的区间值优化问题 (IVOP) 的 LU-解的 KKT 和弱互补近似 KKT(简记为 W-CAKKT) 最优性条件, 其中问题 (IVOP) 的目标区间值函数是弱连续可微的. 首先, 在适当的约束规范下, 证明了 KKT 条件是问题 (IVOP) 存在 LU-解的必要条件. 其次, 引入 W-CAKKT条件, 并证明了在不需要任何约束规范的情况下, W-CAKKT 条件是问题 (IVOP) 存在局部 LU-解的必要条件. 进一步, 在凸性假设下, 证明了 W-CAKKT 条件也是问题 (IVOP) 存在 LU-解的充分条件. 最后, 在满足一定约束规范时, 证明了 W-CAKKT 必要条件优于 KKT 必要条件. 文中的主要结果把一些已有结果从标量优化问题推广到区间值优化问题.

关键词: 区间值优化问题, 互补近似 KKT 条件, LU-解, 约束规范

Abstract:

In this paper, the Karush-Kuhn-Tucker (KKT) and the weakly complementary approximate Karush-Kuhn-Tucker (W-CAKKT) optimality conditions for an LU-solution of the interval-valued optimization problem with inequality and equality constraints (IVOP) are investigated, where the interval-valued objective function is weakly continuously differentiable. Firstly, under suitable constraint qualification, it is proved that the KKT condition is necessary for an LU-solution of (IVOP). Secondly, the W-CAKKT condition is introduced. Absence of any constraint qualification, it is proved that the W-CAKKT condition is necessary for an LU-solution of (IVOP). In addition, under the assumption of convexity, the W-CAKKT condition is sufficient for an LU-solution of (IVOP). Moreover, when some constraint qualification is satisfied, it is shown that the W-CAKKT necessary condition is better than the KKT necessary condition. The results presented in this paper generalize known results from scalar optimization problems to interval-valued optimization problems.

Key words: Interval-Valued optimization problem, Complementary approximate KKT condition, LU-solution, Constraint qualification

中图分类号:

O221

黄晓美, 唐国吉. 区间值优化问题的 KKT 和弱互补近似 KKT 条件[J]. 数学物理学报, 2023, 43(6): 1897-1913.

Huang Xiaomei, Tang Guoji. KKT and Weakly Complementary Approximate KKT Conditions for Interval-Valued Optimization Problems[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1897-1913.

参考文献 22

[1]	Andreani R, Haeser G, Martínez J M. On sequential optimality conditions for smooth constrained optimization. Optimization, 2011, 60(5): 627-641 doi: 10.1080/02331930903578700
[2]	Andreani R, Martínez J M, Svaiter B F. A new sequential optimality condition for constrained optimization and algorithmic consequences. SIAM J Optim, 2010, 20(6): 3533-3554 doi: 10.1137/090777189
[3]	Antczak T. Optimality condition and duality results for nonsmooth vector optimization problems with the multiple interval-valued objective function. Acta Math Sci, 2017, 37B(4): 1133-1150
[4]	Bagirov A, Karmitsa N, MäKelä M M. Introduction to Nonsmooth Optimization. Cham: Springer, 2014
[5]	Bertsekas D P. Nonlinear Programming. Belmont: Athena Scientific, 1999
[6]	Chalco-Cano Y, Lodwick W A, Lizana R. Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative. Fuzzy Optim Decis Ma, 2013, 12(3): 305-322
[7]	Dutta J, Deb K, Tulshyan R, Arora R. Approximate KKT points and a proximity measure for termination. J Glob Optim, 2013, 56(4): 1463-1499 doi: 10.1007/s10898-012-9920-5
[8]	Giorgi G, Jiménez B, Novo V. Approximate Karush-Kuhn-Tucker condition in multiobjective optimization. J Optim Theory Appl, 2016, 171: 70-89 doi: 10.1007/s10957-016-0986-y
[9]	Haeser G, Schuverdt M L. On approximate KKT condition and its extension to continuous variational inequalities. J Optim Theory Appl, 2011, 149(3): 528-539 doi: 10.1007/s10957-011-9802-x
[10]	Hejazi M A. On approximate Karush-Kuhn-Tucker conditions for multiobjective optimization problems. Iran J Sci Technol A, 2018, 42: 873-879
[11]	Ishibuchi H, Tanaka H. Multiobjective programming in optimization of the interval objective function. Eur J Oper Res, 1990, 48(2): 219-225 doi: 10.1016/0377-2217(90)90375-L
[12]	Jayswal A, Stancu-Minasian I, Ahmad I. On sufficiency and duality for a class of interval-valued programming problems. Appl Math Comput, 2011, 218: 4119-4127
[13]	Lai K K, Singh S K, Mishra S K. Approximate-Karush-Kuhn-Tucker conditions and interval valued vector variational inequalities. Wseas Transactions on Mathematics, 2020, 19: 280-288 doi: 10.37394/23206
[14]	Li L F, Liu S Y, Zhang J K. On interval-valued invex mappings and optimality conditions for interval-valued optimization problems. J Inequal Appl, 2015, 2015: Article number 179
[15]	Luu D V, Mai T T. Optimality and duality in constrained interval-valued optimization. 4OR-Q J Oper Res, 2018, 16: 311-337 doi: 10.1007/s10288-017-0369-8
[16]	Moore R E. Method and Applications of Interval Analysis. Philadelphia: SIAM, 1979
[17]	Stancu-Minasian I M, Tigan S. Multiobjective mathematical programming with inexact data. Eur J Oper Res, 1990, 48: 219-225 doi: 10.1016/0377-2217(90)90375-L
[18]	Sun Y H, Wang L S. Optimality conditions and duality in nondifferentiable interval-valued programming. J Ind Manang Optim, 2013, 9: 131-142
[19]	Wu H C. The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur J Oper Res, 2007, 176(1): 46-59 doi: 10.1016/j.ejor.2005.09.007
[20]	Wu H C. On interval-valued nonlinear programming problems. J Math Anal Appl, 2008, 338: 299-316 doi: 10.1016/j.jmaa.2007.05.023
[21]	Zhang J K. Optimality condition and Wolfe duality for invex interval-valued nonlinear programming problems. J Appl Math, 2013, 2013: Article ID 641345
[22]	Zhang J K, Liu S Y, Li L F, Feng Q X. The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function. Optim Lett, 2014, 8: 607-631 doi: 10.1007/s11590-012-0601-6