Jean-Pierre Crouzeix

An Algorithm for Generalized Fractional Programs

An algorithm is suggested that finds the constrained minimum of the maximum of finitely many ratios. The method involves a sequence of linear (convex) subproblems if the ratios are linear (convex-concave). Convergence results as well as rate of convergence results are derived. Special consideration is given to the case of (a) compact feasible regions and (b) linear ratios.

Algorithms for generalized fractional programming

A generalized fractional programming problem is specified as a nonlinear program where a nonlinear function defined as the maximum over several ratios of functions is to be minimized on a feasible domain of ℝn. The purpose of this paper is to outline basic approaches and basic types of algorithms available to deal with this problem and to review their convergence analysis. The conclusion includes results and comments on the numerical efficiency of these algorithms.

Pseudomonotone variational inequality problems: existence of solutions

Necessary and sufficient conditions for the set of solutions of a pseudomonotone variational inequality problem to be nonempty and compact are given.

Characterizations of generalized monotone maps

This paper is a sequel to Ref. 1 in which several kinds of generalized monotonicity were introduced for maps. They were related to generalized convexity properties of functions in the case of gradient maps. In the present paper, we derive first-order characterizations of generalized monotone maps based on a geometrical analysis of generalized monotonicity. These conditions are both necessary and sufficient for generalized monotonicity. Specialized results are obtained for the affine case.

Criteria for quasi-convexity and pseudo-convexity: Relationships and comparisons

A first order criterion for pseudo-convexity and second order criteria for quasi-convexity and pseudo-convexity are given for twice differentiable functions on open convex sets. The relationships between these second order criteria and other known criteria are also analysed. Finally, the numbers of operations required to verify these criteria are calculated and compared.

Duality in generalized linear fractional programming

We consider a generalization of a linear fractional program where the maximum of finitely many linear ratios is to be minimized subject to linear constraints. For this Min-Max problem, a dual in the form of a Max-Min problem is introduced and duality relations are established.

A note on an algorithm for generalized fractional programs

We present a modification of an algorithm recently suggested by the same authors in this journal (Ref. 1). The speed of convergence is improved for the same complexity of computation.

Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities

Abstract.Let VIP(F,C) denote the variational inequality problem associated with the mapping F and the closed convex set C. In this paper we introduce weak conditions on the mapping F that allow the development of a convergent cutting-plane framework for solving VIP(F,C). In the process we introduce, in a natural way, new and useful notions of generalized monotonicity for which first order characterizations are presented.

Additively decomposed quasiconvex funtions

AbstractIn a recently published paper with the same title, Debreu and Koopmans have studied conditions which imply the quasiconvexity of the function