Regularization Methods for Semidefinite Programming

We introduce a new class of algorithms for solving linear semidefinite programming (SDP) problems. Our approach is based on classical tools from convex optimization such as quadratic regularization and augmented Lagrangian techniques. We study the theoretical properties and we show that practical implementations behave very well on some instances of SDP having a large number of constraints. We also show that the “boundary point method” from Povh, Rendl, and Wiegele [Computing, 78 (2006), pp. 277-286] is an instance of this class.

[1]  Jérôme Malick,et al.  A Dual Approach to Semidefinite Least-Squares Problems , 2004, SIAM J. Matrix Anal. Appl..

[2]  Defeng Sun,et al.  Quadratic Convergence and Numerical Experiments of Newton ’ s Method for Computing the Nearest Correlation Matrix , 2005 .

[3]  Franz Rendl,et al.  A Spectral Bundle Method for Semidefinite Programming , 1999, SIAM J. Optim..

[4]  Henry Wolkowicz,et al.  Handbook of Semidefinite Programming , 2000 .

[5]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[6]  B. Martinet.  Brève communication. Régularisation d'inéquations variationnelles par approximations successives , 1970 .

[7]  Renato D. C. Monteiro,et al.  First- and second-order methods for semidefinite programming , 2003, Math. Program..

[8]  Franz Rendl,et al.  An Augmented Primal-Dual Method for Linear Conic Programs , 2008, SIAM J. Optim..

[9]  Franz Rendl,et al.  Semidefinite programming relaxations for graph coloring and maximal clique problems , 2007, Math. Program..

[10]  Jérôme Malik.  Algorithmique en analyse variationnelle et en optimisation semidéfinie , 2005 .

[11]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[12]  Kim-Chuan Toh,et al.  A Newton-CG Augmented Lagrangian Method for Semidefinite Programming , 2010, SIAM J. Optim..

[13]  Kim-Chuan Toh,et al.  Solving Large Scale Semidefinite Programs via an Iterative Solver on the Augmented Systems , 2003, SIAM J. Optim..

[14]  Hans D. Mittelmann,et al.  An independent benchmarking of SDP and SOCP solvers , 2003, Math. Program..

[15]  R. Tyrrell Rockafellar,et al.  Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming , 1976, Math. Oper. Res..

[16]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[17]  Alexander Schrijver,et al.  A comparison of the Delsarte and Lovász bounds , 1979, IEEE Trans. Inf. Theory.

[18]  Renato D. C. Monteiro,et al.  Large-scale semidefinite programming via a saddle point Mirror-Prox algorithm , 2007, Math. Program..

[19]  M. Hestenes.  Multiplier and gradient methods , 1969 .

[20]  Renato D. C. Monteiro,et al.  A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization , 2003, Math. Program..

[21]  Samuel Burer,et al.  Solving Lift-and-Project Relaxations of Binary Integer Programs , 2006, SIAM J. Optim..

[22]  Michael Stingl,et al.  On the solution of large-scale SDP problems by the modified barrier method using iterative solvers , 2009, Math. Program..

[23]  M. J. D. Powell,et al.  A method for nonlinear constraints in minimization problems , 1969 .

[24]  H. Aslaksen.  National university of singapore. , 1995, Environmental science & technology.

[25]  B. V. Dean,et al.  Studies in Linear and Non-Linear Programming. , 1959 .

[26]  M. Trick,et al.  Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, Workshop, October 11-13, 1993 , 1996 .

[27]  KoăźVaraMichal,et al.  On the solution of large-scale SDP problems by the modified barrier method using iterative solvers , 2007 .

[28]  R. Bellman,et al.  Numerical Inversion of the Laplace Transform: Applications to Biology, Economics Engineering, and Physics , 1967 .

[29]  Franz Rendl,et al.  A Boundary Point Method to Solve Semidefinite Programs , 2006, Computing.

[30]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.