Computational enhancements in low-rank semidefinite programming

We discuss computational enhancements for the low-rank semidefinite programming algorithm, including the extension to block semidefinite programs (SDPs), an exact linesearch procedure, and a dynamic rank reduction scheme. A truncated-Newton method is also introduced, and several preconditioning strategies are proposed. Numerical experiments illustrating these enhancements are provided on a wide class of test problems. In particular, the truncated-Newton variant is able to achieve high accuracy in modest amounts of time on maximum-cut-type SDPs.

[1]  K. Shadan,et al.  Available online: , 2012 .

[2]  Jean-Pierre Bourguignon,et al.  Mathematische Annalen , 1893 .

[3]  E. Schmidt.  Zur Theorie der linearen und nichtlinearen Integralgleichungen , 1907 .

[4]  L. Mirsky.  SYMMETRIC GAUGE FUNCTIONS AND UNITARILY INVARIANT NORMS , 1960 .

[5]  G. Golub,et al.  Linear least squares solutions by householder transformations , 1965 .

[6]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[7]  Alexander I. Barvinok,et al.  Problems of distance geometry and convex properties of quadratic maps , 1995, Discret. Comput. Geom..

[8]  Martin P. Bendsøe,et al.  Free material optimization via mathematical programming , 1997, Math. Program..

[9]  Chao Yang,et al.  ARPACK users' guide - solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods , 1998, Software, environments, tools.

[10]  Christian H. Bischof,et al.  A BLAS-3 Version of the QR Factorization with Column Pivoting , 1998, SIAM J. Sci. Comput..

[11]  Gábor Pataki,et al.  On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues , 1998, Math. Oper. Res..

[12]  B. Borchers.  A C library for semidefinite programming , 1999 .

[13]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[14]  B. Borchers.  CSDP, A C library for semidefinite programming , 1999 .

[15]  Brian Borchers,et al.  A library of semidefinite programming test problems , 1999 .

[16]  Stefan Schmieta,et al.  The DIMACS library of semidefinite-quadratic-linear programs , 1999 .

[17]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[18]  Jack Dongarra,et al.  LAPACK Users' Guide, 3rd ed. , 1999 .

[19]  Xiong Zhang,et al.  Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization , 1999, SIAM J. Optim..

[20]  M. Kojima,et al.  Interior-Point Methods for Lagrangian Duals of Semidefinite Programs , 2000 .

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

[22]  Jorge Nocedal,et al.  Automatic Preconditioning by Limited Memory Quasi-Newton Updating , 1999, SIAM J. Optim..

[23]  Yinyu Ye,et al.  Solving Sparse Semidefinite Programs Using the Dual Scaling Algorithm with an Iterative Solver , 2000 .

[24]  Kim-Chuan Toh,et al.  SDPT3 — a Matlab software package for semidefinite-quadratic-linear programming, version 3.0 , 2001 .

[25]  Jorge J. Moré,et al.  Benchmarking optimization software with performance profiles , 2001, Math. Program..

[26]  Yin Zhang,et al.  Digital Object Identifier (DOI) 10.1007/s101070100279 , 2000 .

[27]  Kim-Chuan Toh,et al.  Solving Some Large Scale Semidefinite Programs via the Conjugate Residual Method , 2002, SIAM J. Optim..

[28]  Christoph Helmberg,et al.  A spectral bundle method with bounds , 2002, Math. Program..

[29]  M. Kocvara,et al.  On the modelling and solving of the truss design problem with global stability constraints , 2002 .

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

[31]  Michael Stingl,et al.  PENNON: A code for convex nonlinear and semidefinite programming , 2003, Optim. Methods Softw..

[32]  Daniel Cremers,et al.  Binary Partitioning, Perceptual Grouping, and Restoration with Semidefinite Programming , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[33]  Yin Zhang,et al.  A computational study of a gradient-based log-barrier algorithm for a class of large-scale SDPs , 2003, Math. Program..

[34]  Masakazu Kojima,et al.  Implementation and evaluation of SDPA 6.0 (Semidefinite Programming Algorithm 6.0) , 2003, Optim. Methods Softw..

[35]  Samuel Burer,et al.  Semidefinite Programming in the Space of Partial Positive Semidefinite Matrices , 2003, SIAM J. Optim..

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

[37]  M. Overton,et al.  The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions. , 2004, The Journal of chemical physics.

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

[39]  Andrew A. Kennings,et al.  A semidefinite optimization approach for the single-row layout problem with unequal dimensions , 2005, Discret. Optim..

[40]  Renato D. C. Monteiro,et al.  Digital Object Identifier (DOI) 10.1007/s10107-004-0564-1 , 2004 .

[41]  Michal Kočvara,et al.  On the solution of large-scale SDP problems by the modified barrier method using iterative solvers , 2007, Math. Program..