Michel Volle

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.

A stability result in quasi-convex programming

Combining a result on the lower semicontinuity of the intersection of two convex-valued multifunctions and the level set approach of epi-convergence, we obtain results on the epi-upper semicontinuity of the supremum and the sum of two families of quasi-convex functions. As a consequence, we give some condition ensuring the stability of a quasi-convex program under a perturbation of the objective functions and the constraint sets.

Duality principles for optimization problems dealing with the difference of vector-valued convex mappings

Using the concept of a subdifferential of a vector-valued convex mapping, we provide duality formulas for the minimization of nonconvex composite functions and related optimization problems such as the minimization of a convex function over a vectorial DC constraint.

Quasiconvex Duality for the Max of Two Functions

We provide two formulas on the infimum of the maximum of two functions. Both of them involve quasiconvex or quasiconcave functions and their generalized conjugates. Classical and recent results in convex or nonconvex duality are derived.

Functional Inequalities in the Absence of Convexity and Lower Semicontinuity with Applications to Optimization

In this paper we extend some results in [Dinh, Goberna, Lopez, and Volle, Set-Valued Var. Anal., to appear] to the setting of functional inequalities when the standard assumptions of convexity and lower semicontinuity of the involved mappings are absent. This extension is achieved under certain condition relative to the second conjugate of the involved functions. The main result of this paper, Theorem 1, is applied to derive some subdifferential calculus rules and different generalizations of the Farkas lemma for nonconvex systems, as well as some optimality conditions and duality theory for infinite nonconvex optimization problems. Several examples are given to illustrate the significance of the main results and also to point out the potential of their applications to get various extensions of Farkas-type results and to the study of other classes of problems such as variational inequalities and equilibrium models.

Convex functions with continuous epigraph or continuous level sets

Convex functions with continuous epigraph in the sense of Gale and Klée have been studied recently by Auslender and Coutat in a finite-dimensional setting. Here, we provide characterizations of such functionals in terms of the Legendre-Fenchel transformation in general locally convex spaces. Also, we show that the concept of continuous convex sets is of interest in these spaces. We end with a characterization of convex functions on Euclidean spaces with continuous level sets.

An approximate Hahn–Banach theorem for positively homogeneous functions

This note provides an approximate version of the Hahn–Banach theorem for non-necessarily convex extended-real valued positively homogeneous functions of degree one. Given such a function defined on the real vector space , and a linear function defined on a subspace of and dominated by (i.e. for all ), we say that can approximately be -extended to , if is the pointwise limit of a net of linear functions on , every one of which can be extended to a linear function defined on and dominated by . The main result of this note proves that can approximately be -extended to if and only if is dominated by , the pointwise supremum over the family of all the linear functions on which are dominated by .

Zero Duality Gap for Convex Programs: A Generalization of the Clark–Duffin Theorem

This article uses classical notions of convex analysis over Euclidean spaces, like Gale & Klee’s boundary rays and asymptotes of a convex set, or the inner aperture directions defined by Larman and Brøndsted for the same class of sets, to provide a generalization of the Clark–Duffin Theorem.On this ground, we are able to characterize objective functions and, respectively, feasible sets for which the duality gap is always zero, regardless of the value of the constraints and, respectively, of the objective function.

Optimal boundedness criteria for extended-real-valued functions

The notion of genuinely bounded below function is introduced and characterized by means of the concept of co-equilibrated function. As an application, we state two boundedness criteria for extended-real-valued functions, both optimal in a clearly defined sense. The first one says that an extended-real-valued function minorized by an affine map and coinciding from some value up with a co-equilibrated function is bounded below. The second criterion states that an extended-real-valued function minorized by an affine map is bounded below provided that one of its sub-level sets is co-equilibrated.

Duality For D.C. Optimization Over Compact Sets

Convex duality theory has been successfully used for different optimization problems dealing with differences of convex functions (see for instance [10, 8, 3, 12, 11, 4] and references therein). This paper contains an alternative approach to the duality theory for d.c. programming problems developed in [6]. That theory associates to the general problem of minimizing a d.c. function under several d.c. constraints on an arbitrary locally convex space a dual problem defined in terms of conjugate functions, in such a way that, under suitable constraint qualifications, a strong duality theorem holds. These constraint qualifications are satisfied in specific situations considered in [6]. However they are in general difficult to verify.