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

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....

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Veröffentlicht in:Mathematical programming Jg. 136; H. 2; S. 353 - 373
Hauptverfasser: Grippo, Luigi, Palagi, Laura, Piacentini, Mauro, Piccialli, Veronica, Rinaldi, Giovanni
Format: Journal Article Tagungsbericht
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer-Verlag 01.12.2012
Springer
Springer Nature B.V
Schlagworte:
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
Online-Zugang:Volltext
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Zusammenfassung: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-012-0593-0