## A general asymptotic function with applications

Abstract: Due to its definition through the epigraph, the usual asymptotic function of convex analysis is a very effective tool for studying minimization, especially of a convex function. However, it is not as convenient, if one wants to study maximization of a function “f”; this is done usually through the hypograph or, equivalently, through “−f”. We introduce a new concept of asymptotic function which allows us to simultaneously study convex and concave functions as well as quasi-convex and quasi-concave functions. We provide some calculus rules and relevant...

Read More## Characterizing the calmness property in convex semi-infinite optimization. Modulus estimates

Abstract: We present an overview of the main results on calmness in convex semi-infinite optimization. The first part addresses the calmness of the feasible set and the optimal set mappings for the linear semi-infinite optimization problem in the setting of canonical perturbations, and also in the framework of full perturbations. While there exists a clear proportionality between the calmness moduli of the feasible set mappings in both contexts, the analysis of the relationship between the calmness moduli of the argmin mappings is much more complicated. Point-based expressions (only...

Read More## A VARIATIONAL APPROACH TO SECOND-ORDER OPTIMALITY

ABSTRATC: Conditions associated with local optimality, whether necessary or sufficient, have traditionally been approached through techniques of generalized differentiation. On the first-order level, this has been a long-standing success, although serious challenges remain for equilibrium constraints and the like. On the second-order level, a difficulty areses with the complex concepts of generalized second derivatives and the sometimes-inadequate calculus for determining them. In fact, sufficient second-order conditions of a practical sort, which are the most important aid for numerical...

Read More## Theorems of Borsuk-Ulam Type

Abstract: The Borsuk-Ulam Theorem states that for any continuous function f from S^n to R^n there is some x in S^n such that f(x) = f(-x). Replace S^n by the boundary of some open set A of E=R^{n+1} and replace R^n by some n dimensional manifold B. The conclusion of the theorem remains, with the pair x, -x replaced by some x,y on the boundary whose convex combinations contain some fixed point z in the interior of that open set. Indeed there is a topological structure to all such solutions when the z is considered a variable. If B is not a manifold, the conclusion fails. However if we allow...

Read More## Modeling energy markets with bilevel games

Abstract: Once upon a time, electricity was merely a fairy tale. Since it has been mastered and distributed though, ensuring the supply-demand balance has always been a challenge. Instead of constantly adapting the production to the demand, a new approach consisting in adapting the demand to the production arose about thirty years ago. This approach is called demand-side management, and can be applied through various techniques, notably pricing: offering time-dependent electricity prices can influence the demand. After a small introduction on bilevel programming, we consider a bilevel energy...

Read More## Farkas’ lemma: some extensions and applications

Abstract: The classical Farkas’ lemma characterizing the linear inequalities which are consequence of an ordinary linear system was proved in 1902 by this Hungarian Physicist to justify the first order necessary optimality condition for nonlinear programming problems stated by Ostrogradski in 1838. At present, any result characterizing the containment of the solution set of a given system in the sublevel sets of a given function is said to be a Farkas-type lemma. These results provide partial answers to the so-called containment problem, consisting in characterizing the inclusion of a...

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