On Conjugations for Functions with Values in Extensions of Ordered Groups

We continue our study of generalized conjugations for functions with values in the canonical enlargement of a complete ordered group, started in [10], which encompass various kinds of known conjugations and polarities. We obtain extensions, to this framework, of some results on d.c. duality theory and subdifferentials, and we give some applications to conjugations and subdifferentials for functions with values in .     

Yuan's alternative theorem and the maximization of the minimum eigenvalue function

LetA ₁ andA ₂ be two symmetric matrices of ordern×n. According to Yuan, there exists a convex combination of these matrices which is positive semidefinite, if and only if the functionxR ⁿ max {x ^T A ₁ x,x ^T A ₂ x} is nonnegative. We study the case in which more than two matrices are involved. We study also a related question concerning the maximization of the minimum eigenvalue of a convex combination of symmetric matrices.This research was partially supported by Dirección General de Investigación Científica y Técnica (DGICYT) under Project PB92-0615.     

Characterization of d.c. functions in terms of quasidifferentials

A characterization of d.c. functions f:Ω→R in terms of the quasidifferentials of f is obtained, where Ω is an open convex set in a real Banach space. Recall that f is called d.c. (difference of convex) if it can be represented as a difference of two finite convex functions. The relation of the obtained results with known characterizations is discussed, specifically the ones from [R. Ellaia, A. Hassouni, Characterization of nonsmooth functions through their generalized gradients, Optimization 22 (1991), 401-416] in the finite-dimensional case and [A. Elhilali Alaoui, Caractérisation des fonctions DC, Ann. Sci. Math. Québec 20 (1996), 1-13] in the case of a Banach space.     

On the infimum of a quasiconvex vector function over an intersection

We give sufficient conditions for the infimum of a quasiconvex vector function f over an intersection \(\bigcap_{i\in I}R_{i}\) to agree with the supremum of the infima of f over the R _i ’s.     

Surrogate duality for vector optimization

Using our theorems (of [12]) on separation of convex sets by linear operators, in the sense of the lexi-cographical order on Rⁿ, we prove some theorems of surrogate duality for vector optimization problems with convex constraints (but no regularity assumption), where the surrogate constraint sets are generalized half-spaces and the surrogate multipliers are linear operators, or isomorphisms, or isometries. In the cae of inequality constraints, we prove that the surrogate multipliers can be taken lexicographically non-negative isometries or non-negative (in the usual order) linear isomorphisms.     

The structure of hemispaces in Rn

We study hemispaces (i,e., convex sets with convex complements) in Rⁿ. We give several geometric characterizations of hemispaces and several ways of representing them with the aid of linear operators and lexicographical order. We obtain a metric-affine classification of hemispaces, in terms of their “rank” and “type,” and a “decomposition theorem.” We also give some characterizations of affine transformations which preserve a hemispace.     

Subdifferentials with respect to dualities

LetX andW be two sets and: ¯R^X ¯R^W a duality (i.e., a mapping such that for all and all index setsI). We introduce and study the subdifferential of a function at a pointx _o X, with respect to. We also consider the particular cases when is a (Fenchel-Moreau) conjugation, or a -duality, or a -duality, in the sense of [8].     

Characterizations of evenly convex sets and evenly quasiconvex functions

The aim of this paper is to present a geometric characterization of even convexity in separable Banach spaces, which is not expressed in terms of dual functionals or separation theorems. As an application, an analytic equivalent definition for the class of evenly quasiconvex functions is derived.     

Level sets and the minimal time function of linear control processes

We obtain several theoretical results concerning functions defined from an increasing family of subsets of Rⁿ depending on a real parameter and the level sets of an arbitrary function. These results are applied to the study of the minimal time function of a control process described by a linear differential system, under different assumptions on the control restraint set.