13,452 research outputs found
Some relations between Minty variational-like inequality problems and vectorial optimization problems in Banach spaces
This paper is devoted to the study of relationships between solutions of Stampacchia
and Minty vector variational-like inequalities, weak and strong Pareto solutions of vector
optimization problems and vector critical points in Banach spaces under pseudo-invexity
and pseudo-monotonicity hypotheses. We have extended the results given by Gang and
Liu (2008) [22] to Banach spaces and the relationships obtained for weak efficient points
in Santos et al. (2008) [21] are completed and enabled to relate vector critical points,
weak efficient points, solutions of the Minty and Stampacchia weak vector variationallike inequalities problems and solutions of perturbed vector variational-like inequalities
problems
Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities
This article has two objectives. Firstly, we use the vector variational-like inequalities
problems to achieve local approximate (weakly) efficient solutions of the vector optimization problem
within the novel field of the Hadamard manifolds. Previously, we introduced the concepts of
generalized approximate geodesic convex functions and illustrated them with examples. We see the
minimum requirements under which critical points, solutions of Stampacchia, and Minty weak
variational-like inequalities and local approximate weakly efficient solutions can be identified,
extending previous results from the literature for linear Euclidean spaces. Secondly, we show
an economical application, again using solutions of the variational problems to identify Stackelberg
equilibrium points on Hadamard manifolds and under geodesic convexity assumptions
Generalized vector variational-like inequalities and vector optimization
Abstract In this paper, we consider different kinds of generalized vector variational-like inequality problems and a vector optimization problem. We establish some relationships between the solutions of generalized Minty vector variational-like inequality problem and an efficient solution of a vector optimization problem. We define a perturbed generalized Stampacchia vector variational-like inequality problem and discuss its relation with generalized weak Minty vector variational-like inequality problem. We establish some existence results for solutions of our generalized vector variational-like inequality problems
Nonemptiness and Compactness of Solutions Set for Nondifferentiable Multiobjective Optimization Problems
A nondifferentiable multiobjective optimization problem with nonempty set constraints is considered, and the equivalence of weakly efficient solutions, the critical points for the nondifferentiable multiobjective optimization problems, and solutions for vector variational-like inequalities is established under some suitable conditions. Nonemptiness and compactness of the solutions set for the nondifferentiable multiobjective optimization problems are proved by using the FKKM theorem and a fixed-point theorem
Generalized Stampacchia Vector Variational-Like Inequalities and Vector Optimization Problems Involving Set-Valued Maps
We first obtain that subdifferentials of set-valued mapping from finite-dimensional spaces to finite-dimensional possess certain relaxed compactness. Then using this weak compactness, we establish gap functions for generalized Stampacchia vector variational-like inequalities which are defined by means of subdifferentials. Finally, an existence result of generalized weakly efficient solutions for vector optimization problem involving a subdifferentiable and preinvex set-valued mapping is established by exploiting the existence of a solution for the weak formulation of the generalized Stampacchia vector variational-like inequality via a Fan-KKM lemma
Variational inequalities characterizing weak minimality in set optimization
We introduce the notion of weak minimizer in set optimization. Necessary and
sufficient conditions in terms of scalarized variational inequalities of
Stampacchia and Minty type, respectively, are proved. As an application, we
obtain necessary and sufficient optimality conditions for weak efficiency of
vector optimization in infinite dimensional spaces. A Minty variational
principle in this framework is proved as a corollary of our main result.Comment: Includes an appendix summarizing results which are submitted but not
published at this poin
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