We consider two generalized Minty vector variational-like inequalities and investigate the relations between their solutions and vector optimization problems for non-differentiable α-invex functions.

Generalized Minty vector variational-like inequalities and vector optimization problems in Asplund spaces

In this paper, we establish some results which exhibit an application of convexificators in vector optimization problems (VOPs) and vector variational inequaities involving locally Lipschitz functions. We formulate vector variational inequalities of Stampacchia and Minty type in terms of convexificators and use these vector variational inequalities as a tool to find out necessary and sufficient conditions for a point to be a vector minimal point of the VOP. We also consider the corresponding weak versions of the vector variational inequalities and establish several results to find out weak vector minimal points.

On vector optimization problems and vector variational inequalities using convexificators

A new class of exponential form of vector variational-like inequality problems is introduced, and then the equivalence among (weakly) efficient solutions, vector critical points of vector optimization problem and the solutions of vector variational-like inequalities under the framework of (p,r)-invexity is established. To the best our knowledge, the presented results are new and highly application oriented to other results based on generalized invexities to the context of optimization problems in the literature.

Exponential type vector variational-like inequalities and vector optimization problems with exponential type invexities

In this paper, we introduce Stampacchia-type inverted vector variational inequalities and Minty-type inverted vector variational inequalities and discuss Minty lemma for the inequalities showing the existence of solutions to them in Banach spaces. Next, we consider the equivalence of our Minty lemma with Brouwer’s fixed point theorem as an application.

Minty lemma for inverted vector variational inequalities

In this paper, we formulate vector variational-like inequalities of Stampacchia and Minty type in terms of the limiting subdierentials and use them to solve a nonsmooth V-invex vector optimization problem by utilizing a mean value theorem in Asplund spaces. Under the assumptions of nonsmooth V-invexity, we also derive relationships among the weak formulations of the generalized Minty vector variational-like inequalities, the generalized Stampacchia vector variational-like inequalities and the weak ecient solutions of the nonsmooth V-invex vector optimization problem in Asplund spaces.

On Generalized Minty and Stampacchia Vector Variational-Like Inequalities and V-Invex Vector Optimization in Asplund Spaces

In this paper, the Minty vector variational-like inequality, the Stampacchia vector variational-like inequality, and the weak formulations of these two inequalities defined by means of Mordukhovich limiting subdifferentials are introduced and studied in Asplund spaces. Some relations between the vector variational-like inequalities and vector optimization problems are established by using the properties of Mordukhovich limiting subdifferentials. An existence theorem of solutions for the weak Minty vector variational-like inequality is also given.

Vector variational-like inequalities and vector optimization problems in Asplund spaces

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Vector variational-like inequalities and vector optimization problems in Asplund spaces