Ivan Ginchev

Existence of Solutions and Star-shapedness in Minty Variational Inequalities

Minty Variational Inequalities (for short, Minty VI) have proved to characterize a kind of equilibrium more qualified than Stampacchia Variational Inequalities (for short, Stampacchia VI). This conclusion leads to argue that, when a Minty VI admits a solution and the operator F admits a primitive f (that is F= f′), then f has some regularity property, e.g. convexity or generalized convexity. In this paper we put in terms of the lower Dini directional derivative a problem, referred to as Minty VI(f′_,K), which can be considered a nonlinear extension of the Minty VI with F=f′ (K denotes a subset of ℝn). We investigate, in the case that K is star-shaped, the existence of a solution of Minty VI(f’_,K) and increasing along rays starting at x* property of (for short, F ɛIAR (K,x*)). We prove that Minty VI(f’_,K) with a radially lower semicontinuous function fhas a solution x* ɛker K if and only if FɛIAR(K, x*). Furthermore we investigate, with regard to optimization problems, some properties of increasing along rays functions, which can be considered as extensions of analogous properties holding for convex functions. In particular we show that functions belonging to the class IAR(K,x*) enjoy some well-posedness properties.

First-order optimality conditions in set-valued optimization

A a set-valued optimization problem minC F(x), x ∈X0, is considered, where X0 ⊂ X, X and Y are normed spaces, F: X0 ⊂ Y is a set-valued function and C ⊂ Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x0,y0), y0 ∈F(x0), and are called minimizers. The notions of w-minimizers (weakly efficient points), p-minimizers (properly efficient points) and i-minimizers (isolated minimizers) are introduced and characterized through the so called oriented distance. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive in terms of the Dini directional derivative first order necessary conditions and sufficient conditions a pair (x0, y0) to be a w-minimizer, and similarly to be a i-minimizer. The i-minimizers seem to be a new concept in set-valued optimization. For the case of w-minimizers some comparison with existing results is done.

First-Order Conditions for C0,1 Constrained Vector Optimization

For a Fritz John type vector optimization problem with C0,1 data we define different type of solutions, give their scalar characterizations applying the so called oriented distance, and give necessary and sufficient first order optimality conditions in terms of the Dini derivative. While establishing the sufficiency, we introduce new type of efficient points referred to as isolated minimizers of first order, and show their relation to properly efficient points. More precisely, the obtained necessary conditions are necessary for weakly efficiency, and the sufficient conditions are both sufficient and necessary for a point to be an isolated minimizer of first order.

Second-order conditions in C 1,1 constrained vector optimization

We consider the constrained vector optimization problem minCf(x), g(x) ∈ −K, where f:ℝn→ℝm and g:ℝn→ℝp are C1,1 functions, and Cℝm and Kℝp are closed convex cones with nonempty interiors. Two type of solutions are important for our considerations, namely w-minimizers (weakly efficient points) and i-minimizers (isolated minimizers). We formulate and prove in terms of the Dini directional derivative second-order necessary conditions for a point x0 to be a w-minimizer and second-order sufficient conditions for x0 to be an i-minimizer of order two. We discuss the reversal of the sufficient conditions under suitable constraint qualifications of Kuhn-Tucker type. The obtained results improve the ones in Liu, Neittaanmäki, Křížek [21].

Higher Order Optimality Conditions in Nonsmooth Optimization

For an arbitrary function

Higher order optimality conditions in nonsmooth vector optimization

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Points of Efficiency in Vector Optimization with Increasing-along-rays Property and Minty Variational Inequalities

Abstract The vector optimization problem for the set-valued function is considered, where X 0 is a subset of a real Banach space X and Y is a real Banach space with a given partial order defined by a convex, closed and pointed cone with nonempty interior C. Introducing two infinite elements ±∞C and putting we transform this problem into the equivalent problem for the set-valued function , where p is the extension of F 0 with values +∞C on X\X 0. The nonsmoothness is understood that no regularity conditions for F are preliminary required. Efficient and weakly efficient optimality conditions are reviewed. Higher order lower directional derivatives for F are defined. In terms of these derivatives higher order necessary and sufficient optimality conditions are obtained. The results are illustrated by examples. The concept of an isolated minimizer is introduced. It is shown that the sufficìent conditions characterize (that is they are both sufficient and necessary) the isolated minimizer.

Variational Inequalities in Vector Optimization

In terms of n-th order Dini directional derivative with n positive integer we define n-pseudoconvex functions being a generalization of the usual pseudoconvex functions. Again with the n-th order Dini derivative we define n-stationary points, and prove that a point x 0 is a global minimizer of a n-pseudoconvex function f if and only if x 0 is a n-stationary point of f. Our main result is the following. A radially continuous function f defined on a radially open convex set in a real linear space is n-pseudoconvex if and only if f is quasiconvex function and any n-stationary point is a global minimizer. This statement generalizes the results of Crouzeix, Ferland, Math. Program. 23 (1982), 193–205, and Komlosi, Math. Program. 26 (1983), 232–237. We study also other aspects of the n-pseudoconvex functions, for instance their relations to variational inequalities.

A note on minty type vector variational inequalities

Minty variational inequalities are studied as a tool for vector optimization. Instead of focusing on vector inequalities, we propose an approach through scalarization which allows to construct a proper variational inequality type problem to study any concept of efficiency in vector optimization.