Proper Efficiency for Set-Valued Optimization Problems and Vector Variational-Like Inequalities

Long, Xin; Quan, Jing; Wen, Dao-Jun

doi:10.4134/bkms.2013.50.3.777

Cited by 5 publications

(3 citation statements)

References 23 publications

(19 reference statements)

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“…(iii) Investigations of relations between vector variational inequalities and (set-valued) multi-objective problems can also be found in [12,13,61,92,107,150,153,154,172,182,192,190].…”

Section: Applications In Multi-objective Optimizationmentioning

confidence: 99%

Vector variational inequalities and related topics: A survey of theory and applications

2019

Appl. Set-Valued Anal. Optim.

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In this survey paper, we give a detailed introduction to some of the recent developments in the field of vector variational inequalities and related problems. By giving several examples and presenting the necessary mathematical background and theories, the survey attempts to draw a broad audience and is accessible to students in mathematics and engineering. In doing this, we will study several scalarization methods for vector variational inequalities, which are necessary, sufficient, or both, for the original problem. We further analyze topological and algebraical properties of the solution set. In particular, the existence of solutions is discussed by presenting several existence results. For this purpose, coercivity conditions ensuring, in a certain sense, the boundedness of the data of the vector variational inequality are required. We will further give a precise overview about existence results and techniques, which are known in the literature. Besides that, we analyze a regularization method for non-coercive vector variational inequalities, which consists of approximating solutions of non-coercive problems by a family of regularized vector variational inequalities. After that, we will study relations between vector variational inequalities and multi-objective optimization problems. Motivated by the duality principle in optimization, we also investigate two inverse vector variational inequalities. Furthermore, we consider gap functions for vector variational inequalities, which enable us to study equivalent optimization problems instead. A completely different approach consists of replacing the vector variational inequality by a parametric system or intersection problem. The idea of image space analysis is to study the vector problem in the image space, using one of the previous reformulations. Since vector variational inequalities are ill-posed in general, in the sense that they may either have no solution or multiple solutions, we study stability and sensitivity analysis results. Especially continuity properties of the corresponding solution mapping are investigated. Finally, we give a brief analysis of stochastic vector variational inequalities, generalized problems and numerical methods.

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“…(iii) Investigations of relations between vector variational inequalities and (set-valued) multi-objective problems can also be found in [12,13,61,92,107,150,153,154,172,182,192,190].…”

Section: Applications In Multi-objective Optimizationmentioning

confidence: 99%

Vector variational inequalities and related topics: A survey of theory and applications

2019

Appl. Set-Valued Anal. Optim.

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“…On the other hand, Giannessi [11] initially introduced the notion of vector variational inequality in the finite dimensional Euclidean space. Since then, different types of extensions and generalizations of variational inequalities have been proposed in different settings, see [6,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] and references therein. In [6], optimality conditions are given by a certain kind of vector variational inequality in real normed spaces.…”

Section: Introductionmentioning

confidence: 99%

Globally proper efficiency of set-valued optimization and vector variational inequality involving the generalized contingent epiderivative

Chen

Zhou

2018

J Inequal Appl

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In this paper, firstly, a new property of the cone subpreinvex set-valued map involving the generalized contingent epiderivative is obtained. As an application of this property, a sufficient optimality condition for constrained set-valued optimization problem in the sense of globally proper efficiency is derived. Finally, we establish the relations between the globally proper efficiency of the set-valued optimization problem and the globally proper efficiency of the vector variational inequality.

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“…Accordingly, vector variational inequalities have been generalized [1,13,15] and extended in the context of nonsmooth functions by many researchers [2,7,12,14,17]. The role of generalized invexity is of utmost importance in order to establish the relationship between vector variational-like inequalities and vector optimization problems.…”

mentioning

confidence: 99%

Exponential type vector variational-like inequalities and nonsmooth vector optimization problems

Jayswal

Choudhury

2014

J. Appl. Math. Comput.

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This paper is devoted to study a new class of exponential form of vector variational-like inequality problems comprised of locally Lipschitz functions having exponential type invexities. In the setting of Banach space, we investigate the relationship of nonsmooth exponential type vector variational-like inequality problems with vector optimization problems involving nonsmooth ( p, r )-invex functions. Also, we explore the conditions for solvability of the aforesaid nonsmooth exponential type vector variational-like inequality problems using Fan-KKM Theorem. Moreover, we provide examples to elucidate our results.Keywords Locally Lipschitz function · Nonsmooth vector optimization problems · Nonsmooth exponential type vector variational-like inequalities · (p, r)-invexity · Fan-KKM Theorem Mathematics Subject Classification 90C26 · 90C29 · 49J52 · 58E17 · 58E35

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Proper Efficiency for Set-Valued Optimization Problems and Vector Variational-Like Inequalities

Cited by 5 publications

References 23 publications

Vector variational inequalities and related topics: A survey of theory and applications

Vector variational inequalities and related topics: A survey of theory and applications

Globally proper efficiency of set-valued optimization and vector variational inequality involving the generalized contingent epiderivative

Exponential type vector variational-like inequalities and nonsmooth vector optimization problems

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