Notoriously hard (mixed-)binary QPs: empirical evidence on new completely positive approaches

Abstract

By now, many copositive reformulations of mixed-binary QPs have been discussed, triggered by Burer’s seminal characterization from 2009. In conic optimization, it is very common to use approximation hierarchies based on positive-semidefinite (psd) matrices where the order increases with the level of the approximation. Our purpose is to keep the psd matrix orders relatively small to avoid memory size problems in interior point solvers. Based upon on a recent discussion on various variants of completely positive reformulations and their relaxations (Bomze et al. in Math Program 166(1–2):159–184, 2017), we here present a small study of the notoriously hard multidimensional quadratic knapsack problem and quadratic assignment problem. Our observations add some empirical evidence on performance differences among the above mentioned variants. We also propose an alternative approach using penalization of various classes of (aggregated) constraints, along with some theoretical convergence analysis. This approach is in some sense similar in spirit to the alternating projection method proposed in Burer (Math Program Comput 2:1–19, 2010) which completely avoids SDPs, but for which no convergence proof is available yet.