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In the first chapter of this book the basic results within convex and quasiconvex analysis are presented. In Section 2 we consider in detail the algebraic and topological properties of convex sets within ℝ together with their primal and dual representations. In Section 3 we apply the results for convex sets to convex and quasiconvex functions and show how these results can be used to give primal and dual representations of the functions considered in this field. As such, most of the results are well known with the exception of Subsection 3.4 dealing with dual representations of quasiconvex functions. In Section 3 we consider applications of convex analysis to noncooperative game and minimax theory, Lagrangian duality in optimization and the properties of positively homogeneous evenly quasiconvex functions. Among these result an elementary proof of the well-known Sion’s minimax theorem concerning quasiconvex-quasiconcave bifunctions is presented, thereby avoiding the less elementary fixed point arguments. Most of the results are proved in detail and the authors have tried to make these proofs as transparent as possible. Remember that convex analysis deals with the study of convex cones and convex sets and these objects are generalizations of linear subspaces and affine sets, thereby extending the field of linear algebra. Although some of the proofs are technical, it is possible to give a clear geometrical interpretation of the main ideas of convex analysis. Finally in Section 5 we list a short and probably incomplete overview on the history of convex and quasiconvex analysis.

This chapter is devoted to first and second order characterizations of quasi/pseudo convexity of a function and first order characterizations of quasi/pseudo monotonicity of a single-valued map. Some applications are given.

The convexity of the epigraph of a convex function induces important properties with respect to the continuity and differentiability of the function. Moreover, the function is locally Lipschitz in the interior of the domain of the function. If for a quasiconvex function, the convexity concerns the lower level sets and not the epigraph, some important properties on continuity and differentiability are still preserved. An important property of quasiconvex functions is that they are locally nondecreasing with respect to some positive cone.

In this chapter, the role of generalized convex functions in optimization is stressed. A particular attention is devoted to local-global properties, to optimality of stationary points and to sufficiency of first order necessary optimality conditions for scalar and vector problems. Despite of the numerous classes of generalized convex functions suggested in these last fifty years, we have limited ourselves to introduce and study those classes of scalar and vector functions which are more used in the literature.

In this chapter we introduce the notion of convexity and generalized convexity including invexity for vector valued functions. Some characterizations of these functions are provided. Then we study vector problems involving generalized convex functions. The major aspects of this study concern the existence of efficient solutions, optimality conditions using contingent derivatives and approximate Jacobians, scalarization for convex and quasiconvex problems, and topological properties of efficient solution sets of generalized convex problems.

This article presents an approach to generalized convex duality theory based on Fenchel-Moreau conjugations; in particular, it discusses quasiconvex conjugation and duality in detail. It also describes the related topic of microeconomics duality and analyzes the monotonicity of demand functions.

In this paper we study the emerging area of abstract convexity. The theory of abstract convex functions and sets arises out of the properties of convex functions related to their global nature. One of the main applications of abstract convexity is global optimization. Apart from discussing the various fundamental facts about abstract convexity we also study quasiconvex functions in the light of abstract convexity. We further describe the surprising applications of the ideas of abstract convexity to the study of Hadamard type inequalities for quasiconvex functions.

Single-ratio and multi-ratio fractional programs in applications are often generalized convex programs. We begin with a survey of applications of single-ratio fractional programs, min-max fractional programs and sum- of-ratios fractional programs. Given the limited advances for the latter class of problems, we focus on an analysis of min-max fractional programs. A parametric approach is employed to develop both theoretical and algorithmic results.

We first present nine kinds of (generalized) monotone maps and in case of gradient maps their counterpart of nine kinds of (generalized) convex functions. In addition we present topologically pseudomonotone maps. We then derive sufficient and/or necessary conditions for various kinds of generalized monotonicity for several subclasses of maps. We study differentiable maps, locally Lipschitz maps, general continuous maps and affine maps.

This chapter is devoted to the study of nonsmooth generalized convex functions with the help of special classes of generalized derivatives. Several results are presented on the links between generalized monotonicity of the generalized derivatives and generalized convexity of the functions under discussion. The abundance of the different notions of generalized derivatives has motivated an axiomatic treatment resulting, among others, in the concept of first order approximation. The usefulness of quasiconvex first order approximations in optimization theory is investigated, in particular, generalized upper quasidifferentiable functions are studied, quasiconvex Farkas Theorems and KKT-type optimality conditions are elaborated.