TY - JOUR

T1 - SpeeDP

T2 - An algorithm to compute SDP bounds for very large Max-Cut instances

AU - Grippo, Luigi

AU - Palagi, Laura

AU - Piacentini, Mauro

AU - Piccialli, Veronica

AU - Rinaldi, Giovanni

PY - 2012/12

Y1 - 2012/12

N2 - 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.

AB - 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.

KW - Low rank factorization

KW - Max-Cut

KW - Nonlinear programming

KW - Semidefinite programming

KW - Unconstrained binary quadratic programming

UR - http://www.scopus.com/inward/record.url?scp=84870368819&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84870368819&partnerID=8YFLogxK

U2 - 10.1007/s10107-012-0593-0

DO - 10.1007/s10107-012-0593-0

M3 - Article

AN - SCOPUS:84870368819

VL - 136

SP - 353

EP - 373

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 2

ER -