Recent Results on Duality in Vector Optimization

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra or the union of finitely many generalized polyhedra. Dropping the compactness assumption, we establish some results on structure of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set, and Pareto optimal value set of (SVOP) and on connectedness of Pareto solution set and Pareto optimal value set of (SVOP). In particular, we improved and generalize, Arrow, Barankin, and Blackwell’s classical results in Euclidean spaces and Zheng and Yang’s results in general Banach spaces.

Download Full-text

Vector variational inequalities and their relations with vector optimization

An International Journal of Optimization and Control Theories & Applications (IJOCTA) ◽

10.11121/ijocta.01.2014.00160 ◽

2013 ◽

Vol 4 (1) ◽

pp. 35-44 ◽

Cited By ~ 2

Author(s):

Surjeet Kaur Suneja ◽

Bhawna Kohli

Keyword(s):

Variational Inequalities ◽

Vector Optimization ◽

Optimization Problem ◽

Vector Optimization Problem ◽

Clarke Subdifferential ◽

Vector Variational Inequalities ◽

Regularity Assumption ◽

Weak Minimum ◽

Minimum Solution ◽

Clarke Subdifferentials

In this paper, K- quasiconvex, K- pseudoconvex and other related functions have been introduced in terms of their Clarke subdifferentials, where is an arbitrary closed convex, pointed cone with nonempty interior. The (strict, weakly) -pseudomonotonicity, (strict) K- naturally quasimonotonicity and K- quasimonotonicity of Clarke subdifferential maps have also been defined. Further, we introduce Minty weak (MVVIP) and Stampacchia weak (SVVIP) vector variational inequalities over arbitrary cones. Under regularity assumption, we have proved that a weak minimum solution of vector optimization problem (VOP) is a solution of (SVVIP) and under the condition of K- pseudoconvexity we have obtained the converse for MVVIP (SVVIP). In the end we study the interrelations between these with the help of strict K-naturally quasimonotonicity of Clarke subdifferential map.

Download Full-text

Derivatives of Hadamard type in vector optimization

Mathematica Slovaca ◽

10.1515/ms-2017-0113 ◽

2018 ◽

Vol 68 (2) ◽

pp. 421-430

Author(s):

Karel Pastor

Keyword(s):

Nonlinear Analysis ◽

Optimality Conditions ◽

Vector Optimization ◽

Optimality Condition ◽

Optimization Problem ◽

Vector Optimization Problem ◽

Dini Derivatives ◽

Scalar Optimality ◽

Derivatives Of

Abstract In our paper we will continue the comparison which was started by Vsevolod I. Ivanov [Nonlinear Analysis 125 (2015), 270–289], where he compared scalar optimality conditions stated in terms of Hadamard derivatives for arbitrary functions and those which was stated for ℓ-stable functions in terms of Dini derivatives. We will study the vector optimization problem and we show that also in this case the optimality condition stated in terms of Hadamard derivatives is more advantageous.

Download Full-text