Lifts of convex sets in optimization

Volker Kaibel, Rekha Thomas
2015 Mathematical programming  
This special issue is dedicated to the geometry and complexity of lifts or extended formulations of convex sets which has been an active area of research in recent years. This developing field lies at the intersection of several areas such as convex geometry, polyhedral theory, real algebraic geometry, combinatorics, optimization, and computer science, among others. The idea of finding efficient representations of convex sets (especially polytopes) by expressing them as projections of simple
more » ... vex sets in higher dimensions is not new and is well-known in areas such as integer programming and real algebraic geometry. The current view of the topic is inspired by the 1991 paper of Mihalis Yannakakis who showed that the matching polytope does not admit a small "symmetric" polyhedral lift, in striking contrast to the classical result of Jack Edmonds that linear optimization over the matching polytope is possible in polynomial time. Yannakakis also showed that there is a precise connection between the nonnegative rank of the slack matrix of a polytope and the size of the smallest polyhedral lift that is possible for this polytope. Both of these results have been extended in multiple directions in recent papers. The restrictions imposed by symmetry on the size of lifts has been explored in multiple papers by now. Also, the connection between lifts and nonnegative ranks have been extended to general convex sets and closed cones. In particular, positive semidefinite lifts of polytopes have received a fair bit of attention lately, while the older notion of B Volker Kaibel
doi:10.1007/s10107-015-0940-z fatcat:dojppko74nhpzjvuzbvp5gerlu