Representability of convex sets by analytical linear inequality systems

Non-representability of finite projective planes by convex sets

We prove that there is no d such that all finite projective planes can be represented by convex sets in R, answering a question of Alon, Kalai, Matoušek, and Meshulam. Here, if P is a projective plane with lines l1, . . . , ln, a representation of P by convex sets in R is a collection of convex sets C1, . . . , Cn ⊆ R d such that Ci1 , Ci2 . . . , Cik have a common point if and only if the corr...

Mixed-Integer Convex Representability

We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer assignments is finite. We develop a characterization for the more general case of unbounded integer variables together with a simple necessary condition for re...

2 Linear Matrix Inequality Representation of Sets

This article concerns the question: which subsets of R can be represented with Linear Matrix Inequalities, LMIs? This gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory. Also before having much hope of representing engineering problems as LMIs by automatic methods one needs a good idea of which problems can and cannot be re...

2 Linear Matrix Inequality Representation of Sets

Quantitative Stability of Linear Infinite Inequality Systems Under Block Perturbations With Applications to Convex Systems

The original motivation for this paper was to provide an efficient . quantitative analysis of convex infinite.(or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bo...