Semidefinite Relaxations for Integer Programming

Abstract

We survey some recent developments in the area of semidefinite optimization applied to integer programming. After recalling some generic modeling techniques to obtain semidefinite relaxations for NP-hard problems, we look at the theoretical power of semidefinite optimization in the context of the Max-Cut and the Coloring Problem. In the second part, we consider algorithmic questions related to semidefinite optimization, and point to some recent ideas to handle large scale problems. The survey is concluded with some more advanced modeling techniques, based on matrix relaxations leading to copositive matrices.

Identity Matrix

“…Other possible relaxations include Lagrangian relaxations (Fisher 1981;Geoffrion 1974), semi-definite programming relaxations (Rendl 2010), and combinatorial relaxations, e.g., the one-tree relaxation for the traveling salesman problem Held and Karp (1970). This discussion initially considers use of the LP relaxation, since this is the simplest one and the one used in state-of-the-art software.…”

Obstacles, Challenges and Strategies for Facies Estimation in AVO Seismic Inversion

“…Linear relaxations of these problems have long been known to produce frustratingly poor lower bounds for the optimum. Newer semidefinite programming (SDP) lifting schemes for BQP problems (Rendl, 2010) have provided remarkably good bounds for canonical problems like maxcut: integrality gaps of 5% are widely reported. These methods work in the convex space of positive-definite matrices X "lifted" from ( 1 F )( 1 F T ) with objectiveF = argmax Tr(−QX ) subject to X 0, and general linear constraints on elements of X .…”

A guide to conic optimisation and its applications

“…In fact, it is possible to construct entire hierarchies of SDP relaxations; see, e.g., [56,59,65,85,89]. Moreover, SDP has been successfully applied to many 0-1 quadratic programs; see [44,60,80,91] for early work on the subject, and [51,95] for some recent applications.…”