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

Luigi Grippo, Laura Palagi, Mauro Piacentini, Veronica Piccialli, Giovanni Rinaldi

Research output: Contribution to journalArticlepeer-review

Abstract

We consider low-rank semidefinite programming (LRSDP) relaxations of unconstrained \{-1,1\} 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.

Original languageEnglish
Pages (from-to)353-373
Number of pages21
JournalMathematical Programming
Volume136
Issue number2
DOIs
Publication statusPublished - Dec 2012

Keywords

  • Low rank factorization
  • Max-Cut
  • Nonlinear programming
  • Semidefinite programming
  • Unconstrained binary quadratic programming

ASJC Scopus subject areas

  • Mathematics(all)
  • Software

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