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Stronger Kuhn-Tucker type conditions in nonsmooth multiobjective optimization: Locally Lipschitz case

JOURNAL OF optimization THEORY AND APPLICATIONS 2005年第2期127卷 367-388页

作者： Li, XF Zhang, JZ Jilin Univ Dept Math Changchun Peoples R China City Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China

For an inequality constrained nonsmooth multiobjective optimization problem, where the objective and constraint functions are locally Lipschitz, a nonsmooth analogue of the Maeda-type Guignard constraint qualification is given;stronger Kuhn-Tucker type necessary optimality conditions are derived that are expressed in terms of upper convexificators. Moreover, other constraint qualifications sufficient for the nonsmooth analogue are introduced and their relationships are presented.

关键词： nonsmooth multiobjective optimization constraint qualifications stronger Kuhn-Tucker conditions locally Lipschitz functions directional Dini derivatives convexificators

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Constraint qualifications in nonsmooth multiobjective optimization

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JOURNAL OF optimization THEORY AND APPLICATIONS 2000年第2期106卷 373-398页

作者： Li, XF Jilin Univ Technol Dept Math Appl Changchun Peoples R China City Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China

For an inequality constrained nonsmooth multiobjective optimization problem involving locally Lipschitz functions, stronger KT-type necessary conditions and KT necessary conditions (which in the continuously differentiable case reduce respectively to the stronger KT conditions studied recently by Maeda and the usual KT conditions) are derived for efficiency and weak efficiency under several constraint qualifications. Stimulated by the stronger KT-type conditions, the notion of core of the convex hull of the union of finitely many convex sets is introduced. As main tool in the derivation of the necessary conditions, a theorem of the alternatives and a core separation theorem are also developed which are respectively extensions of the Motzkin transposition theorem and the Tucker theorem.

关键词： nonsmooth multiobjective optimization constraint qualifications KT necessary conditions Lipschitz continuity Clarke subdifferential

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Optimality and duality in nonsmooth multiobjective optimization involving generalized type I functions

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2003年第10-11期45卷 1497-1506页

作者： Kuk, H Tanino, T Kyung Hee Univ Dept Math Sch Elect & Informat Yongin 449701 South Korea Osaka Univ Grad Sch Engn Dept Elect & Informat Syst Suita Osaka 5650871 Japan

A nonsmooth multiobjective optimization problem involving generalized Type I vector-valued functions is considered. Karush-Kuhn-Tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be an efficient or properly efficient solution. Duality theorems are proved for Wolfe type and Mond-Weir type duals under the generalized Type I assumptions. (C) 2003 Elsevier Science Ltd. All rights reserved.

关键词： nonsmooth multiobjective optimization type I functions optimality duality

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Optimality and duality for a nonsmooth multiobjective optimization involving generalized type I functions

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2008年第3期67卷 493-504页

作者： Mishra, S. K. Wang, S. Y. Lai, K. K. Chinese Acad Sci Acad Math & Syst Sci Inst Syst Sci Beijing 100080 Peoples R China Govind Ballabh Pant Univ Agr & Technol Coll Basic Sci & Humanities Dept Math Stat & Comp Sci Pantnagar 263145 Uttar Pradesh India City Univ Hong Kong Dept Management Sci Kowloon Hong Kong Peoples R China

A nonsmooth multiobjective optimization problem involving generalized (F, alpha, rho, d)-type I function is considered. Karush-Kuhn-Tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be an efficient or properly efficient solution. Duality results are obtained for mixed type dual under the aforesaid assumptions.

关键词： nonsmooth multiobjective optimization efficient solution type I functions generalized convexity mixed duality

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On Tucker-Type Alternative Theorems and Necessary Optimality Conditions for nonsmooth multiobjective optimization

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JOURNAL OF optimization THEORY AND APPLICATIONS 2022年第2期195卷 480-503页

作者： Feng, Min Li, Shengjie Wang, Jie Chongqing Jiaotong Univ Coll Math & Stat Chongqing 400074 Peoples R China Chongqing Univ Coll Math & Stat Chongqing 401331 Peoples R China

This paper concentrates on necessary conditions for properly efficient solutions in nonsmooth multiobjective optimization problems. We first present a generalization of Tucker's alternative theorem for conic nonlinear systems, provided that a closedness condition holds. Some sufficient conditions for the validity of such a closedness condition are given. As applications, under the weak Abadie regularity condition, we then establish the primal and the strong Karush/Kuhn-Tucker (dual) necessary optimality conditions for an efficient solution to be locally properly efficient in Borwein's sense. The primal and the dual conditions are formulated as an equivalent pair by means of the Tucker-type alternative results. Finally we give an example to illustrate that Borwein's locally properly efficient solution cannot be reduced to the only efficient one in the main results.

关键词： nonsmooth multiobjective optimization Theorems of the alternative Properly efficient solutions Weak regularity conditions Strong Karush Kuhn-Tucker conditions

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Existence and Boundedness of the Kuhn-Tucker Multipliers in nonsmooth multiobjective optimization

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JOURNAL OF optimization THEORY AND APPLICATIONS 2010年第2期145卷 373-386页

作者： Li, X. F. Zhang, J. Z. Jilin Univ Dept Appl Math Sch Math Sci Changchun 130023 Peoples R China BJNU HKBU United Int Coll Zhuhai Peoples R China

Using the idea of upper convexificators, we propose constraint qualifications and study existence and boundedness of the Kuhn-Tucker multipliers for a nonsmooth multiobjective optimization problem with inequality constraints and an arbitrary set constraint. We show that, at locally weak efficient solutions where the objective and constraint functions are locally Lipschitz, the constraint qualifications are necessary and sufficient conditions for the Kuhn-Tucker multiplier sets to be nonempty and bounded under certain semiregularity assumptions on the upper convexificators of the functions.

关键词： Upper convexificators Constraint qualifications Existence and boundedness of Kuhn-Tucker multipliers nonsmooth multiobjective optimization

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An interactive method for nonsmooth multiobjective optimization with an application to optimal control

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optimization Methods and Software 1993年第1期2卷 31-44页

An implementable method for nonsmooth multiobjective optimization is described. The algorithm is a modification of the well-known Geoffrion-Dyer-Feinberg (GDF) method for smooth interactive multiobjective problems. The smooth gradient-based Frank-Wolfe method exploited in the GDF method is replaced by a modified (Kiev) subgradient method in order to compute the search direction. The solutions are projected onto the set of Pareto optimal points by using exact penalty scalarizing functions. A bundle-type method is utilized to solve the nonsmooth single objective optimization problems arising in every iteration of the *** an application we introduce a model of an elastic string, which leads us to solve a nonsmooth multiobjective optimal control problem governed by a variational inequality. Due to the unilateral boundary conditions, the state of the system depends in a nonsmooth way on the control variable. Finally, some encouraging numerical experience is reported.

关键词： nonsmooth multiobjective optimization Geoffrion-Dyer-Feinberg method bundle methods optimal control

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Optimality conditions for nonsmooth multiobjective bilevel optimization using tangential subdifferentials

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RAIRO-OPERATIONS RESEARCH 2021年第5期55卷 3041-3048页

作者： Jennane, Mohsine Kalmoun, El Mostafa El Fadil, Lhoussain Sidi Mohamed Ben Abdellah Univ Dept Math FSDM Fes Morocco Al Akhawayn Univ Ifrane Sch Sci & Engn POB 104 Ifrane 53000 Morocco

In combining the value function approach and tangential subdifferentials, we establish necessary optimality conditions of a nonsmooth multiobjective bilevel programming problem under a suitable constraint qualification. The upper level objectives and constraint functions are neither assumed to be necessarily locally Lipschitz nor convex.

关键词： nonsmooth multiobjective optimization bilevel programming optimality conditions optimal value function constraint qualifications tangential subdifferentials

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First- and second-order optimality conditions of nonsmooth sparsity multiobjective optimization via variational analysis

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JOURNAL OF GLOBAL optimization 2024年第2期89卷 303-325页

作者： Chen, Jiawei Su, Huasheng Ou, Xiaoqing Lv, Yibing Southwest Univ Sch Math & Stat Chongqing 400715 Peoples R China Chongqing Coll Humanities Sci & Technol Sch Management Chongqing 401524 Peoples R China Yangtze Univ Sch Informat & Math Jingzhou 434023 Peoples R China

In this paper, we investigate optimality conditions of nonsmooth sparsity multiobjective optimization problem (shortly, SMOP) by the advanced variational analysis. We present the variational analysis characterizations, such as tangent cones, normal cones, dual cones and second-order tangent set, of the sparse set, and give the relationships among the sparse set and its tangent cones and second-order tangent set. The first-order necessary conditions for local weakly Pareto efficient solution of SMOP are established under some suitable conditions. We also obtain the equivalence between basic feasible point and stationary point defined by the Frechet normal cone of SMOP. The sufficient optimality conditions of SMOP are derived under the pseudoconvexity. Moreover, the second-order necessary and sufficient optimality conditions of SMOP are established by the Dini directional derivatives of the objective function and the Bouligand tangent cone and second-order tangent set of the sparse set.

关键词： nonsmooth multiobjective optimization Sparsity Optimality conditions Second-order tangent set Variational analysis

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Necessary conditions for weak minima and for strict minima of order two in nonsmooth constrained multiobjective optimization

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JOURNAL OF GLOBAL optimization 2021年第1期80卷 177-193页

作者： Constantin, Elena Univ Pittsburgh Math Dept Johnstown PA 15904 USA

In this paper, we give necessary conditions for the existence of a strict local minimum of order two for multiobjective optimization problems with equality and inequality constraints. We suppose that the objective function and the active inequality constraints are only locally Lipschitz. We consider both regular equality constraints and degenerate equality constraints. This article could be considered as a continuation of [E. Constantin, Necessary Conditions for Weak Efficiency for nonsmooth Degenerate multiobjective optimization Problems, J. Global Optim, 75, 111-129, 2019]. We introduce a constraint qualification and a regularity condition, and we show that under each of them, the dual necessary conditions for a weak local minimum of the aforementioned article become of Kuhn-Tucker type.

关键词： Weak local minimum Strict local minimum of order two nonsmooth multiobjective optimization Degenerate equality constraints Locally Lipschitz optimization problems Kuhn-Tucker dual necessary optimality conditions

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