Interior Point and Semidefinite Approaches in Combinatorial Optimization

Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient primal-dual interior-point methods (IPMs) and various first order approaches for the solution of large scale SDPs. This survey presents an up to date account of semidefinite and interior point approaches in solving NP-hard combinatorial optimization problems to optimality, and also in developing approximation algorithms for some of them. The interior point approaches discussed in the survey have been applied directly to non-convex formulations of IPs; they appear in a cutting plane framework to solving IPs, and finally as a subroutine to solving SDP relaxations of IPs. The surveyed approaches include non-convex potential reduction methods, interior point cutting plane methods, primal-dual IPMs and first-order algorithms for solving SDPs, branch and cut approaches based on SDP relaxations of IPs, approximation algorithms based on SDP formulations, and finally methods employing successive convex approximations of the underlying combinatorial optimization problem.