Alberto Zaffaroni

Degrees of Efficiency and Degrees of Minimality

In this work we characterize different types of solutions of a vector optimization problem by means of a scalarization procedure. Usually different scalarizing functions are used in order to obtain the various solutions of the vector problem. Here we consider different kinds of solutions of the same scalarized problem. Our results allow us to establish a parallelism between the solutions of the scalarized problem and the various efficient frontiers: stronger solution concepts of the scalar problem correspond to more restrictive notions of efficiency. Besides the usual notions of weakly efficient and efficient points, which are characterized as global and strict global solutions of the scalarized problem, we also consider some restricted notions of efficiency, such as strict and proper efficiency, which are characterized as Tikhonov well-posed minima and sharp minima for the scalarized problem.

On the notion of proper efficiency in vector optimization

In this paper, we consider the main definitions of proper efficiency for a vector optimization problem in topological linear spaces. The implications among these definitions generalize the inclusion structure holding in Euclidean spaces with componentwise ordering.

Asymptotic conditions for weak and proper optimality in infinite dimensional convex vector optimization

In this paper, we establish necessary and sufficient dual conditions for weak and proper minimality of infinite dimensional vector convex programming problems without any regularity conditions. The optimality conditions are given in asymptotic forms using epigraphs of conjugate functions and sub differentials, It is shown how these asymptotic conditions yield standard Lagrangian conditions under appropriate regularity conditions. The main tool, used to obtain these results, is a new solvability result of Motzkin type for cone convex systems. We also provide local Lagrangian necessary conditions for certain non—convex problems using convex approximations

Is every radiant function the sum of quasiconvex functions

Abstract.An open question in the study of quasiconvex function is the characterization of the class of functions which are sum of quasiconvex functions. In this paper we restrict our attention to quasiconvex radiant functions, i.e. those whose level sets are radiant as well as convex and deal with the claim that a function can be expressed as the sum of quasiconvex radiant functions if and only if it is radiant. Our study is carried out in the framework of Abstract Convex Analysis: the main tool is the description of a supremal generator of the set of radiant functions, i.e. a class of elementary functions whose sup-envelope gives radiant functions, and of the relation between the elementary generators of radiant functions and those of quasiconvex radiant functions. An important intermediate result is a nonlinear separation theorem in which a superlinear function is used to separate a point from a closed radiant set.

Superlinear separation for radiant and coradiant sets

The article studies radiant and coradiant sets of some normed space X from the point of view of separation properties between a set A⊆ X and a point x ∉ A; indeed they show striking similarities with the ones holding for convex sets and can be obtained by simply changing halfspaces (level sets of linear continuous functions), with level sets of continuous superlinear functions. In a geometric perspective one can say that radiant sets are separated by means of convex coradiant sets and coradiant sets are separated by means of convex radiant sets. The identification between the geometric and the analytic approach passes through the well-known Minkoski gauge and the study of concave continuous gauges of convex coradiant sets. The results are then applied to the study of abstract convexity with respect to the family L of continuous superlinear functions, to the characterization of evenly coradiant convex sets and to the subdifferentiability of positively homogeneous functions. Dedicated to D. Pallaschke on his 65th birthday.

Continuous Approximations, Codifferentiable Functions and Minimization Methods

Our starting point relies on the observation that, for a nondifferentiable function, the classical (Gâteaux) directional derivative fails to be continuous with respect to the initial point; this is also related to the lack of continuity properties of the quasidifferential or other differential objects obtained as linearizations of the directional derivative. In this paper we describe the notion of codifferentiability as a mean to obtain a continuous approximation for a nonsmooth function. Particular emphasis is given to applications to optimization theory: necessary optimality conditions, minimization methods, extensions of the Newton method for a system of nonsmooth equations.

Quasiconcavity of Sets and Connectedness of the Efficient Frontier in Ordered Vector Spaces

We introduce new notions of quasiconcavity of sets in ordered vector spaces, extending the properties of sets which are images of convex sets by quasiconcave functions. This allows us to generalize known results and obtain new ones on the connectedness of the sets of various types of efficient solutions.

On the existence of maximal elements for partial preorders

Abstract The aim of this paper is to present a general approach to find conditions which ensure the existence of maximal elements in a partially preordered set. We generalize some known results and establish new ones; moreover we show that our conditions extend recent results in the economic literature and in the theory of vector optimization, which hold under more specific assumptions on the topological and algebraic structure of the space.

Vector subdifferentials via recession mappings ∗ ∗this research was supported by grants from M.U.R.S.T (Itall) and the australian research council.

A vector subdifferential is defined for a class of directionally differentiable mappings between ordered topological vector spaces. The method used to derive the subdifferential is based on the existcnce of a recession mapping for a positively homogeneous operator. The properties of the recession mapping are discussed and they are shown to he similar to those in the real–valued case. In addition a calculus for the vector subdifferential is developed. Final1y these results are used to develop first order necessary optimality conditions for a class of vector optimization problems involving either proper or weak minimality concepts.