Approximations and improvements to semidefinite relaxations of optimization problems

Abstract

At the beginning of the 1990s, semidefinite programming (SDP) gained popularity as one of the major research areas of convex optimization and attracted many researchers from various disciplines. Today many engineers and scientists around the world use semidefinite programming to solve problems in engineering, geometry, combinatorial optimization, polynomial optimization and natural science.

Semidefinite programming provides strong approximation algorithms for many different combinatorial optimization problems including MAXCUT, MAX2SAT and MAX3SAT. We examine the performance guarantee ratios for the above combinatorial optimization problems when using approximate solutions to the SDP relaxations.

Theoretically, numerous algorithms have been introduced to solve SDP problems. But computational costs for solving the problems become quite expensive as the size of the matrix increases. Second order cone programming (SOCP) can be used in cutting plane framework to reduce the computational cost of solving SDP problems. We examine different ways to pick the SOCP cuts to produce better and faster convergence to the solution of the SDP relaxation of the MAXCUT problem.

Finally, we examine polynomial optimization. In particular we examine the application of the Sum of Squares (SOS) method.The standard approach is to replace the global nonnegativity constraint from the original polynomial optimization problem with the SOS constraint. Then, we can turn the SOS problem into as SDP. However, there maybe a gap between the optimal values of the original polynomial optimization problem and the SOS problem. We investigate closing this gap by adding cutting plane acquired using the Arithmetic/Geometric mean inequality.