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

• Published in: Mathematical programming Vol. 136; no. 2; pp. 353 - 373
• Main Authors: Grippo, Luigi, Palagi, Laura, Piacentini, Mauro, Piccialli, Veronica, Rinaldi, Giovanni
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.12.2012; Springer; Springer Nature B.V
• Subjects: Algorithms; Applied sciences; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Equivalence; Exact sciences and technology; Full Length Paper; Graphs; Heuristic; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Nonlinear programming; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Optimization; Programming; Studies; Theoretical; Semidefinite programming; 20G40; 20C20; Low rank factorization; Unconstrained binary quadratic programming; 20E28; Nonlinear programming; Max-Cut; Non convex programming; Semidefinite relaxation; Constraint satisfaction; Semi definite programming; Matrix factorization; Critical point; Non linear programming; Separation method; Zero one programming; Mathematical programming; Unconstrained optimization; Minimization; Quadratic programming; Cutting plane method; Equality constraint; Heuristic method; Representation theory; Quadratic function
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-012-0593-0

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.