Rita Pini

Invexity and generalized convexity

Let f be a real differentiate function defined on the open subset A of R n;f is said to be invex if the following inequality is satisfied for a suitable function η The purpose of this paper is to investigate the relationship between invexity and generalized convexity; moreover, we shall give necessary and sufficient conditions for a differentiable function to be η-invex, which are natural extensions of the analogous conditions for convex functions.

A survey of recent [1985--1995] advances in generalized convexity with applications to duality theory and optimality conditions

Convexity plays a very important role in various branches of mathematical, natural and social sciences. In an effort to extend existing results depending on convexity there has been a steady interest over the years towards its generalizations. Many excellent books, monographs and numerous articles have been written pursuing generalizations of the basic concepts, their characterizations. and studying properties under different conditions. The purpose of this contribution is to gather information on certain generalizations of convexity and their applications to duality theory and optimality conditions

A note on stability for parametric equilibrium problems

In this paper we consider equilibrium problems in vector metric spaces where the function f and the set K are perturbed by the parameters @e,@h. We study the stability of the solutions, providing some results in the peculiar framework of generalized monotone functions, first in the particular case where K is fixed, then under both data perturbation.

A Note on Equilibrium Problems with Properly Quasimonotone Bifunctions

In this paper, we consider some well-known equilibrium problems and their duals in a topological Hausdorff vector space X for a bifunction F defined on K x K,where K is a convex subset of X. Some necessary conditions are investigated, proving different results depending on the behaviour of F on the diagonal set. The concept of proper quasimonotonicity for bifunctions is defined, and the relationship with generalized monotonicity is investigated. The main result proves that the condition of proper quasimonotonicity is sharp in order to solve the dual equilibrium problem on every convex set.

Sensitivity for parametric vector equilibria

In this article, we consider a parametric vector equilibrium problem in topological vector spaces, or metric spaces, if needed, defined as follows: given , find such that where the order in Y is defined by a suitable fixed cone C. We study the upper stability of the map of the solutions S=S(λ), providing results in the peculiar framework of generalized monotone functions. In the particular case of a single-valued solution map, we provide conditions for the Hölder regularity of S in both cases when K is fixed, and also when it depends on a parameter.

Well-posedness for vector equilibrium problems

We introduce and study two notions of well-posedness for vector equilibrium problems in topological vector spaces; they arise from the well-posedness concepts previously introduced by the same authors in the scalar case, and provide an extension of similar definitions for vector optimization problems. The first notion is linked to the behaviour of suitable maximizing sequences, while the second one is defined in terms of Hausdorff convergence of the map of approximate solutions. In this paper we compare them, and, in a concave setting, we give sufficient conditions on the data in order to guarantee well-posedness. Our results extend similar results established for vector optimization problems known in the literature.

Generalized convexity and generalized monotonocity

Abstract General definitions of monotonocity, quasimonotonocity, pseudomonotonocity and strong monotonocity of a vector valued function defined on a subset of an n-dimensional Euclidean space are given. Some properties and characterizations are studied in the genera] setting. A relationship between generalized monotonocity and (Φ1, Φ2)– convexity introduced earlier by the authors, is described under certain conditions. Applications of the general definitions are studied in terms of invexity and B-vexity.

Lexicographic and sequential equilibrium problems

The aim of this work is to analyze lexicographic equilibrium problems on a topological Hausdorff vector space X, and their relationship with some other vector equilibrium problems. Existence results for the tangled lexicographic problem are proved via the study of a related sequential problem. This approach was already followed by the same authors in the case of variational inequalities.

Invariance properties of generalized monotonicity

Generalized monotone maps are studied under affine variable transformations. The results enable us to generate generalized monotone matrices of any size. Various necessary conditions and sufficient conditions for generalized monotone matrices are derived. Furthermore. admissible translations of generalized monotone linear maps are studied. Finally, the maximal domain of generalized monotonicity is characterized.

Lexicographic variational inequalities with applications

In this article, we consider equivalence properties between various kinds of lexicographic variational inequalities. By employing various concepts of monotonicity, we show that the usual sequential variational inequality is equivalent to the direct lexicographic variational inequality or to the dual lexicographic variational inequality. We establish several existence results for lexicographic variational inequalities. Also, we introduce the lexicographic complementarity problem and establish its equivalence with the lexicographic variational inequality. We illustrate our approach by several examples of applications to vector transportation and vector spatial equilibrium problems.