## 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...

Read More## Hidden Convexity in the l0 Pseudonorm and Lower Bound Convex Programs for Exact Sparse Optimization

Abstract: In sparse optimization problems, one looks for solution that have few nonzero components. We consider problems where sparsity is exactly measured by the l0 pseudonorm. We display a suitable conjugacy for which we show that the l0 pseudonorm is equal to its biconjugate. As a corollary, we obtain that the (nonconvex) l0 pseudonorm coincides, on the sphere, with a convex lsc function that we characterize. With this conjugacy, we display a lower bound for the original exact sparse optimization problem, which is a convex minimization program over the unit ball of a...

Read More## Progressive Decoupling of Linkages in Optimization with Elicitable Convexity

ABSTRACT: A method called the Progressive Decoupling Algorithm is described for solving variational inequalities and optimization problems in which a subspace captures “linkages” that can be relaxed. The approach is inspired by the Progressive Hedging Algorithm in convex stochastic programming and resembles the Partial Inverse Method of Spingarn, but retains more parametric flexibility than the latter. It is able even to work when mononicity or convexity is not directly present but can be “elicited”. The role of elicitation mimics the role of...

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