Matteo Rocca

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].

A Characterization of C^(k,1) Functions

In this work we provide a characterization of C{k,1} functions on Rn (that is k times differentiable with locally Lipschitzian k-th derivatives) by means of (k+1)-th divided differences and Riemann derivatives. In particular we prove that the class of C{k,1} functions is equivalent to the class of functions with bounded (k+1)-th divided difference. From this result we deduce aTaylors formula for this class of functions and a characterization through Riemann derivatives.

A survey on C 1,1 fuctions: theory, numerical methods and applications

In this paper we survey some notions of generalized derivative for C 1,1 functions. Furthermore some optimality conditions and numerical methods for nonlinear minimization problems involving C1,1 data are studied.

Some Remarks on Second-order Generalized Derivatives for C wedge(1,1) Functions

Many definitions of second-order generalized derivatives have been introduced to obtain optimality conditions for optimization problems with C wedge(1,1) data. The aim of this note is to show some relations among these definitions.

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

Variational Inequalities in Vector Optimization

In this paper we investigate the links among generalized scalar variational inequalities of differential type, vector variational inequalities and vector optimization problems. The considered scalar variational inequalities are obtained through a nonlinear scalarization by means of the so called ”oriented distance” function [14, 15]. In the case of Stampacchia-type variational inequalities, the solutions of the proposed ones coincide with the solutions of the vector variational inequalities introduced by Giannessi [8]. For Minty-type variational inequalities, analogous coincidence happens under convexity hypotheses. Furthermore, the considered variational inequalities reveal useful in filling a gap between scalar and vector variational inequalities. Namely, in the scalar case Minty variational inequalities of differential type represent a sufficient optimality condition without additional assumptions, while in the vector case the convexity hypothesis was needed. Moreover it is shown that vector functions admitting a solution of the proposed Minty variational inequality enjoy some well-posedness properties, analogously to the scalar case [4].

Minty Variational Inequality and Optimization: Scalar and Vector Case

Minty variational inequalities are considered as related to the scalar minimization problem in which the objective function is a primitive of the operator involved in the inequality itself. Well-posedness (in the sense of Tykhonov) of this primitive problem is proved as a consequence of the existence of a strict solution of a Minty variational inequality.