SpeeDP: an algorithm to compute SDP bounds for very large Max-Cut instances

Abstract

We consider low-rank semidefinite programming (LRSDP) relaxations of unconstrained quadratic problems (or, equivalently, of Max-Cut problems) that can be formulated as the non-convex nonlinear programming problem of minimizing a quadratic function subject to separable quadratic equality constraints. We prove the equivalence of the LRSDP problem with the unconstrained minimization of a new merit function and we define an efficient and globally convergent algorithm, called SpeeDP, for finding critical points of the LRSDP problem. We provide evidence of the effectiveness of SpeeDP by comparing it with other existing codes on an extended set of instances of the Max-Cut problem. We further include SpeeDP within a simply modified version of the Goemans-Williamson algorithm and we show that the corresponding heuristic SpeeDP-MC can generate high-quality cuts for very large, sparse graphs of size up to a million nodes in a few hours

A Coherent Ising Machine for MAX-CUT Problems: Performance Evaluation against Semidefinite Programming and Simulated Annealing

“…For graphs with both positive and negative edge weights, the SDP problem is commonly solved using the interior-point method, which scales as Õ(N 3.5 ) = O(N 3.5 log(1/ε)) [39]. Besides, low rank formulation of SDP is effective when the graph is sparse [40][41][42]. In our computational experiments, the COPL SDP based on the interior point method was used as a solver for MAX-CUT problems [43].…”

Big data algorithms beyond machine learning

“…Recent approaches based on SDP relaxations can solve sparse instances of up to a million nodes in a few hours [26]. Also those approaches can benefit from an initial data reduction, as long as the data reduction is efficient.…”

A nonmonotone GRASP

“…Goemans and Williamson [53] proposed a randomized algorithm that uses semidefinite programming to achieve a performance guarantee of 0.87856 if the weights are nonnegative. Since then, many approximation algorithms for NP-hard problems have been devised using SDP relaxations [56,66,91].…”