A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming

We present a unified analysis for a class of long-step primal-dual path-following algorithms for semidefinite programming whose search directions are obtained through linearization of the symmetrized equation of the central pathHP(XS) ≡ [PXSP−1 + (PXSP−1)T]/2 = μI, introduced by Zhang. At an iterate (X,S), we choose a scaling matrixP from the class of nonsingular matricesP such thatPXSP−1 is symmetric. This class of matrices includes the three well-known choices, namely:P = S1/2 andP = X−1/2 proposed by Monteiro, and the matrixP corresponding to the Nesterov—Todd direction. We show that within the class of algorithms studied in this paper, the one based on the Nesterov—Todd direction has the lowest possible iteration-complexity bound that can provably be derived from our analysis. More specifically, its iteration-complexity bound is of the same order as that of the corresponding long-step primal-dual path-following algorithm for linear programming introduced by Kojima, Mizuno and Yoshise. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

[1]  Stephen P. Boyd,et al.  A primal—dual potential reduction method for problems involving matrix inequalities , 1995, Math. Program..

[2]  Shuzhong Zhang,et al.  On the long-step path-following method for semidefinite programming , 1998, Oper. Res. Lett..

[3]  Shuzhong Zhang,et al.  Symmetric primal-dual path-following algorithms for semidefinite programming , 1999 .

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

[5]  M. Kojima,et al.  A note on the Nesterov-Todd and the Kojima-Shindoh-hara search directions in semidefinite programming , 1999 .

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

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

[8]  F. Jarre.  An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices , 1993 .

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

[10]  Roy E. Marsten,et al.  The interior-point method for linear programming , 1992, IEEE Software.

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

[12]  MASAYUKI SHIDA,et al.  Existence and Uniqueness of Search Directions in Interior-Point Algorithms for the SDP and the Monotone SDLCP , 1998, SIAM J. Optim..

[13]  Yinyu Ye,et al.  A Class of Projective Transformations for Linear Programming , 1990, SIAM J. Comput..

[14]  Kim-Chuan Toh,et al.  On the Nesterov-Todd Direction in Semidefinite Programming , 1998, SIAM J. Optim..

[15]  Michael J. Todd,et al.  Self-Scaled Barriers and Interior-Point Methods for Convex Programming , 1997, Math. Oper. Res..

[16]  M. Shida,et al.  Existence of Search Directions in Interior-Point Algorithms for the SDP and the Monotone SDLCP , 1996 .

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

[18]  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..

[19]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

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

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

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