A Newton-CG Augmented Lagrangian Method for Semidefinite Programming

We consider a Newton-CG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresponding solution mapping at the origin. For the inner problems, we show that the positive definiteness of the generalized Hessian of the objective function in these inner problems, a key property for ensuring the efficiency of using an inexact semismooth Newton-CG method to solve the inner problems, is equivalent to the constraint nondegeneracy of the corresponding dual problems. Numerical experiments on a variety of large-scale SDP problems with the matrix dimension $n$ up to $4,110$ and the number of equality constraints $m$ up to $2,156,544$ show that the proposed method is very efficient. We are also able to solve the SDP problem fap36 (with $n=4,110$ and $m=1,154,467$) in the Seventh DIMACS Implementation Challenge much more accurately than in previous attempts.

[1]  E. H. Zarantonello.  Projections on Convex Sets in Hilbert Space and Spectral Theory: Part I. Projections on Convex Sets: Part II. Spectral Theory , 1971 .

[2]  R. Tyrrell Rockafellar,et al.  A dual approach to solving nonlinear programming problems by unconstrained optimization , 1973, Math. Program..

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

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

[5]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[6]  T. Steihaug.  The Conjugate Gradient Method and Trust Regions in Large Scale Optimization , 1983 .

[7]  S. M. Robinson.  Local structure of feasible sets in nonlinear programming , 1983 .

[8]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[9]  S. M. Robinson.  Local Structure of Feasible Sets in Nonlinear Programming - Part II. Nondegeneracy , 1984 .

[10]  P. Strevens.  Iii , 1985 .

[11]  R. Rockafellar.  Conjugate Duality and Optimization , 1987 .

[12]  S. M. Robinson.  Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity , 1987 .

[13]  B. Kummer.  NEWTON's METHOD FOR NON-DIFFERENTIABLE FUNCTIONS , 1988, Advances in Mathematical Optimization.

[14]  Liqun Qi,et al.  A nonsmooth version of Newton's method , 1993, Math. Program..

[15]  Francisco Facchinei,et al.  Minimization of SC1 functions and the Maratos effect , 1995, Oper. Res. Lett..

[16]  J. Pang,et al.  A globally convergent Newton method for convex SC1 minimization problems , 1995 .

[17]  Michael L. Overton,et al.  Complementarity and nondegeneracy in semidefinite programming , 1997, Math. Program..

[18]  Defeng Sun,et al.  Semismooth Matrix-Valued Functions , 2002, Math. Oper. Res..

[19]  Martin Grötschel,et al.  Frequency planning and ramifications of coloring , 2002, Discuss. Math. Graph Theory.

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

[21]  Panos M. Pardalos,et al.  Finding maximum independent sets in graphs arising from coding theory , 2002, SAC '02.

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

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

[24]  Christoph Helmberg,et al.  Numerical evaluation of SBmethod , 2003, Math. Program..

[25]  Defeng Sun,et al.  Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems , 2003, Math. Oper. Res..

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

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

[28]  Defeng Sun,et al.  Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization , 2005, Math. Program..

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

[30]  Defeng Sun,et al.  A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix , 2006, SIAM J. Matrix Anal. Appl..

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

[32]  Defeng Sun,et al.  The Strong Second-Order Sufficient Condition and Constraint Nondegeneracy in Nonlinear Semidefinite Programming and Their Implications , 2006, Math. Oper. Res..

[33]  Defeng Sun,et al.  A Dual Optimization Approach to Inverse Quadratic Eigenvalue Problems with Partial Eigenstructure , 2007, SIAM J. Sci. Comput..

[34]  Defeng Sun,et al.  The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming , 2008, Math. Program..

[35]  Kim-Chuan Toh,et al.  An inexact primal–dual path following algorithm for convex quadratic SDP , 2007, Math. Program..

[36]  Defeng Sun,et al.  Constraint Nondegeneracy, Strong Regularity, and Nonsingularity in Semidefinite Programming , 2008, SIAM J. Optim..

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

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

[39]  Franz Rendl,et al.  Regularization Methods for Semidefinite Programming , 2009, SIAM J. Optim..

[40]  Franz Rendl,et al.  Copositive and semidefinite relaxations of the quadratic assignment problem , 2009, Discret. Optim..

[41]  Philipp Birken,et al.  Numerical Linear Algebra , 2011, Encyclopedia of Parallel Computing.

[42]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.