Well-posed Vector Optimization Problems and Vector Variational Inequalities

In this paper we introduce notions of well-posedness for a vector optimization problem and for a vector variational inequality of differential type, we study their basic properties and we establish the links among them. The proposed concept of well-posedness for a vector optimization problem generalizes the notion of well-setness for scalar optimization problems, introduced in [2]. On the other side, the introduced definition of well-posedness for a vector variational inequality extends the one given in [13] for the scalar case.Keywords: vector optimization, vector variational inequality, well-posedness

A Minty variational principle for set optimization

Extremal problems are studied involving an objective function with values in (order) complete lattices of sets generated by so called set relations. Contrary to the popular paradigm in vector optimization, the solution concept for such problems, introduced by F. Heyde and A. L\"ohne, comprises the attainment of the infimum as well as a minimality property. The main result is a Minty type variational inequality for set optimization problems which provides a sufficient optimality condition under lower semicontinuity assumptions and a necessary condition under appropriate generalized convexity assumptions. The variational inequality is based on a new Dini directional derivative for set-valued functions which is defined in terms of a "lattice difference quotient": A residual operation in a lattice of sets replaces the inverse addition in linear spaces. Relationships to families of scalar problems are pointed out and used for proofs: The appearance of improper scalarizations poses a major difficulty which is dealt with by extending known scalar results such as Diewert's theorem to improper functions

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].

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.vector optimization, nonsmooth optimization, C0,1 functions, Dini derivatives, first-order optimality conditions, lagrange multipliers