Juan Enrique Martínez-Legaz

Quasiconvex duality theory by generalized conjugation methods

We survey duality theories for quasiconvex optimization problems, based on notions of generalized conjugation. Some of them are obtained from Moreaus generalized conjugation, while the others are special cases of the so-called H-duality. The relationship existing between the two kinds of approaches is described.

Generalized convexity, generalized monotonicity : recent results

Preface. Part I: Generalized Convexity. 1. Are Generalized Derivatives Useful for Generalized Convex Functions? J.-P. Penot. 2. Stochastic Programs with Chance Constraints: Generalized Convexity and Approximation Issues R.J.-B. Wets. 3. Error Bounds for Convex Inequality Systems A.S. Lewis, Jong-Shi Pang. 4. Applying Generalised Convexity Notions to Jets A. Eberhard, et al. 5. Quasiconvexity via Two Step Functions A.M. Rubinov, B.M. Glover. 6. On Limiting Frechet epsilon-Subdifferentials A. Jourani, M. Thera. 7. Convexity Space with Respect to a Given Set L. Blaga, L. Lupsa. 8. A Convexity Condition for the Nonexistence of Some Antiproximinal Sets in the Space of Integrable Functions A.-M. Precupanu. 9. Characterizations of rho-Convex Functions M. Castellani, M. Pappalardo. Part II: Generalized Monotonicity. 10. Characterizations of Generalized Convexity and Generalized Monotonicity, a Survey J.-P. Crouzeix. 11. Quasimonotonicity and Pseudomonotonicity in Variational Inequalities and Equilibrium Problems N. Hadjisavvas, S. Schaible. 12. On the Scalarization of Pseudoconcavity and Pseudomonotonicity Concepts for Vector Valued Functions R. Cambini, S. Komlosi. 13. Variational Inequalities and Pseudomonotone Functions: Some Characterizations R. John. Part III: Optimality Conditions and Duality. 14. Simplified Global Optimality Conditions in Generalized Conjugation Theory F. Flores-Bazan, J.-E. Martinez-Legaz. 15. Duality in DC Programming B. Lemaire, M.Volle. 16. Recent Developments in Second Order Necessary Optimality Conditions A. Cambini, et al. 17. Higher Order Invexity and Duality in Mathematical Programming B. Mond, J. Zhang. 18. Fenchel Duality in Generalized Fractional Programming C.R. Bector, et al. Part IV: Vector Optimization. 19. The Notion of Invexity in Vector Optimization: Smooth and Nonsmooth Case G. Giorgi, A. Guerraggio. 20. Quasiconcavity of Sets and Connectedness of the Efficient Frontier in Ordered Vector Spaces E. Molho, A. Zaffaroni. 21. Multiobjective Quadratic Problem: Characterization of the Efficient Points A. Beato-Moreno, et al. 22. Generalized Concavity for Bicriteria Functions R. Cambini. 23. Generalized Concavity in Multiobjective Programming A. Cambini, L. Martein.

Downward Sets and their separation and approximation properties

We develop a theory of downward subsets of the space ℝI, where I is a finite index set. Downward sets arise as the set of all solutions of a system of inequalities x∈ℝI,ft(x)≤0 (t∈T), where T is an arbitrary index set and each ft (t∈T) is an increasing function defined on ℝI. These sets play an important role in some parts of mathematical economics and game theory. We examine some functions related to a downward set (the distance to this set and the plus-Minkowski gauge of this set, which we introduce here) and study lattices of closed downward sets and of corresponding distance functions. We discuss two kinds of duality for downward sets, based on multiplicative and additive min-type functions, respectively, and corresponding separation properties, and we give some characterizations of best approximations by downward sets. Some links between the multiplicative and additive cases are established.

On lower subdifferentiable functions

This paper studies the notion of lower subdifferentiability from the view point of generalized conjugation. By this method the main properties of lower sub-differentiable functions are analysed. A related conjugation theory for Holder and Lipschitz functions is developed, which is used to characterize those functions which can be expressed as a supremum of Holder functions. Some applications to quasiconvex optimization and optimal control theory are examined.

Duality between direct and indirect utility functions under minimal hypotheses

Abstract We give a characterization of those functions which can be obtained as the indirect utility function associated with the utility function of a consumer. This permits to formulate the duality between direct and indirect utility functions in the most general possible setting, which exhibits a perfect symmetry.

A formula on the approximate subdifferential of the difference of convex functions

We give a formula on the e −subdifferential of the difference of two convex functions. As a by-product of this formula, one recovers a recent result of Hiriart-Urruty, namely, a necessary and sufficient condition for global optimality in nonconvex optimisation.

Dualities between complete lattices

We study dualities between two complete lattices Eand Fi.e., mappings △:E→ F satisfying for all {x i } ieI ⊆E and all index sets I including the empty set I = O. We give characterizations and representations of dualities △, and some results on the dual △* F→Eof △ and on the associated hull operator △*△:E→Ein the general case and in various particular eases. Among several applications, we devote special attention to Fenchel-Moreau conjugations.

Generalized Convex Duality and its Economic Applicatons

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.

Exact quasiconvex conjugation

In this article we develop a conjugacy theory in quasiconvex analysis, in which no lower semicontinuity or normality assumption is needed to ensure the coincidence of the second conjugate of any function with its quasivonvex hull. This is made by an extension of the concept ofH-conjugation, and is based on a separation theorem by general halfspaces.The theory is applied in mathematical programming to define dual problems, which consist in maximizing a quasiconcave function of matricial variable, the optimum being always attained. The absence of duality gap is equivalent to the quasiconvexity of the perturbation function at the origin. A Lagrangian for general problems is studied and compared with the one of Luenberger in the case of vertical perturbations.ZusammenfassungIn der vorliegenden Arbeit wird die Konjugierte einer quasikonvexen Funktion eingeführt, ohne dabei Halbstetigkeit oder Normalität vorauszusetzen, um übereinstimmung der zweiten Konjugierten mit der quasikonvexen Hüllenfunktion zu garantieren. Dies wird durch eine Verallgemeinerung der sogenanntenH-Konjugation erreicht und basiert auf einem Trennungssatz für allgemeine Halbräume.Die Theorie wird dazu herangezogen, ein duales Programm einzuführen, welches ein quasikonkaves Maximumproblem ist, das eine Matrix als Variable hat und stets eine optimale Lösung besitzt. Die Abwesenheit einer Dualitätslücke ist damit gleichbedeutend, daß die Störfunktion im Ursprung quasikonvex ist. Es wird eine Lagrangefunktion für allgemeine Probleme eingeführt und mit derjenigen Luenbergers im Falle vertikaler Störungen verglichen.

Monotonic analysis over cones: II

In this article, we study the class of increasing and convex along rays (ICAR) functions over a cone. Apart from studying its basic properties, we study them from the point of view of Abstract Convexity. Further, we study the relation between the ICAR and Lipschitz functions and the properties under which an ICAR function has a Lipschitz behaviour. We also study the class of decreasing and convex along rays functions (DCAR).