Alternating direction augmented Lagrangian methods for semidefinite programming

• Published in: Mathematical programming computation Vol. 2; no. 3-4; pp. 203 - 230
• Main Authors: Wen, Zaiwen, Goldfarb, Donald, Yin, Wotao
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.12.2010
• Subjects: Full Length Paper; Mathematics; Mathematics and Statistics; Mathematics of Computing; Operations Research/Decision Theory; Optimization; Theory of Computation; Alternating direction method; Augmented Lagrangian method; Semidefinite programming; 90C30; 90C35; 90C22; 90C06
• ISSN: 1867-2949; 1867-2957
• DOI: 10.1007/s12532-010-0017-1

We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration, our basic algorithm minimizes the augmented Lagrangian function for the dual SDP problem sequentially, first with respect to the dual variables corresponding to the linear constraints, and then with respect to the dual slack variables, while in each minimization keeping the other variables fixed, and then finally it updates the Lagrange multipliers (i.e., primal variables). Convergence is proved by using a fixed-point argument. For SDPs with inequality constraints and positivity constraints, our algorithm is extended to separately minimize the dual augmented Lagrangian function over four sets of variables. Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that our algorithms are robust and very efficient due to their ability or exploit special structures, such as sparsity and constraint orthogonality in these problems.