Dualities in Convex Algebraic Geometry

Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article compares three notions of duality that are relevant in these contexts: duality of convex bodies, duality of projective varieties, and the Karush-Kuhn-Tucker conditions derived from Lagrange duality. We show that the optimal value of a polynomial program is an algebraic function whose minimal polynomial is expressed by the hypersurface projectively dual to the constraint set. We give an exposition of recent results on the boundary structure of the convex hull of a compact variety, we contrast this to Lasserre's representation as a spectrahedral shadow, and we explore the geometric underpinnings of semidefinite programming duality.Comment: 48 pages, 11 figure

Duality Results for Conic Convex Programming

This paper presents a unified study of duality properties for the problem of minimizing a linear function over the intersection of an affine space with a convex cone in finite dimension. Existing duality results are carefully surveyed and some new duality properties are established. Examples are given to illustrate these new properties. The topics covered in this paper include Gordon-Stiemke type theorems, Farkas type theorems, perfect duality, Slater condition, regularization, Ramana's duality, and approximate dualities. The dual representations of various convex sets, convex cones and conic convex programs are also discussed.semidefinite programming;duality;conic convex programming

Asymptotic duality over closed convex sets

AbstractThe asymptotic duality theory of linear programming over closed convex cones [4] is extended to closed convex sets, by embedding such sets in appropriate cones. Applications to convex programming and to approximation theory are given

Fenchel and Lagrange Duality are Equivalent

A basic result in ordinary (Lagrange) convex programming is the saddlepoint duality theorem concerning optimization problems with convex inequalities and linear-affine equalities satisfying a Slater condition. This note shows that this result is equivalent to the duality theorem of Fenchel.Supported in part by the U.S. Army Research Office (Durham) under Contract No. DAHC04-73-C-0032

A Comparison of Mixed-Integer Programming Models for Non-Convex Piecewise Linear Cost Minimization Problems

We study a generic minimization problem with separable non-convex piecewise linear costs, showing that the linear programming (LP) relaxation of three textbook mixed integer programming formulations each approximates the cost function by its lower convex envelope. We also show a relationship between this result and classical Lagrangian duality theory

Matching for Teams.

We are given a list of tasks Z and a population divided into several groups X j of equal size. Performing one task z requires constituting a team with exactly one member x j from every group. There is a cost (or reward) for participation: if type x j chooses task z, he receives p j (z); utilities are quasi-linear. One seeks an equilibrium price, that is, a price system that distributes all the agents into distinct teams. We prove existence of equilibria and fully characterize them as solutions to some convex optimization problems. The main mathematical tools are convex duality and mass transportation theory. Uniqueness and purity of equilibria are discussed. We will also give an alternative linear-programming formulation as in the recent work of Chiappori et al.Matching; Equilibria; Convex duality; Optimal transportation;