On the Local Convergence of a Predictor-Corrector Method for Semidefinite Programming

We study the local convergence of a predictor-corrector algorithm for semidefinite programming problems based on the Monteiro--Zhang unified direction whose polynomial convergence was recently established by Monteiro. Under strict complementarity and nondegeneracy assumptions superlinear convergence with Q-order 1.5 is proved if the scaling matrices in the corrector step have bounded condition number. A version of the predictor-corrector algorithm enjoys quadratic convergence if the scaling matrices in both predictor and corrector steps have bounded condition numbers. The latter results apply in particular to algorithms using the Alizadeh--Haeberly--Overton (AHO) direction since there the scaling matrix is the identity matrix.

[1]  R. Vanderbei,et al.  An Interior-point Method for Semideenite Programming an Interior-point Method for Semideenite Programming , 1994 .

[2]  M. Overton,et al.  A New Primal-Dual Interior-Point Method for Semidefinite Programming , 1994 .

[3]  Stephen J. Wright,et al.  Local convergence of interior-point algorithms for degenerate monotone LCP , 1994, Comput. Optim. Appl..

[4]  Shinji Hara,et al.  Interior Point Methods for the Monotone Linear Complementarity Problem in Symmetric Matrices , 1995 .

[5]  Robert J. Vanderbei,et al.  An Interior-Point Method for Semidefinite Programming , 1996, SIAM J. Optim..

[6]  F. Potra,et al.  Superlinear Convergence of a Predictor-corrector Method for Semideenite Programming without Shrinking Central Path Neighborhood , 1996 .

[7]  Renato D. C. Monteiro,et al.  Primal-Dual Path-Following Algorithms for Semidefinite Programming , 1997, SIAM J. Optim..

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

[9]  Shinji Hara,et al.  Interior-Point Methods for the Monotone Semidefinite Linear Complementarity Problem in Symmetric Matrices , 1997, SIAM J. Optim..

[10]  Michael L. Overton,et al.  Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results , 1998, SIAM J. Optim..

[11]  Michael J. Todd,et al.  Primal-Dual Interior-Point Methods for Self-Scaled Cones , 1998, SIAM J. Optim..

[12]  Renato D. C. Monteiro,et al.  Polynomial Convergence of Primal-Dual Algorithms for Semidefinite Programming Based on the Monteiro and Zhang Family of Directions , 1998, SIAM J. Optim..

[13]  Masakazu Kojima,et al.  Local convergence of predictor—corrector infeasible-interior-point algorithms for SDPs and SDLCPs , 1998, Math. Program..

[14]  Yin Zhang,et al.  On Extending Some Primal-Dual Interior-Point Algorithms From Linear Programming to Semidefinite Programming , 1998, SIAM J. Optim..

[15]  Zhi-Quan Luo,et al.  Superlinear Convergence of a Symmetric Primal-Dual Path Following Algorithm for Semidefinite Programming , 1998, SIAM J. Optim..

[16]  F. Potra,et al.  Superlinear Convergence of Interior-Point Algorithms for Semidefinite Programming , 1998 .

[17]  Florian A. Potra,et al.  A Superlinearly Convergent Primal-Dual Infeasible-Interior-Point Algorithm for Semidefinite Programming , 1998, SIAM J. Optim..

[18]  Masakazu Kojima,et al.  A Predictor-corrector Interior-point Algorithm for the Semidenite Linear Complementarity Problem Using the Alizadeh-haeberly-overton Search Direction , 1996 .

[19]  F. Potra,et al.  On a general class of interior-point algorithms for semidefinite programming with polynomial complexity and superlinear convergence , 1999 .