A Blueprint for Semidefinite Relaxations of Binary-Constrained Quadratic Programs Computing tight bounds on NP-hard problems using ADMM

Abstract

This thesis looks at the solution techniques of two NP-hard, large scale problems, the quadratic assignment problem, QAP, and the side chain positioning, SCP, problem. We summarize existing approaches from and look at the two problems in a unified way using a binary-constrained quadratic program, BCQP. We show how to obtain upper and lower bounds for the BCQP by formulating the semidefinite programming (SDP) relaxation and applying the Alternating Direction Method of Multipliers (ADMM) algorithm to solve it. By unifying the two problems under the umbrella of the BCQP, we better understand why the method is so successful for these two problems and obtain a blueprint for applying ADMM to similar combinatorial optimization problems.