Search Directions And Convergence Analysis Of Some Infeasible Path-Following Methods For The Monoton

We consider a family of primal/primal-dual/dual search directions for the monotone LCP over the space of n n symmetric block-diagonal matrices. We consider two infeasible predictor-corrector path-following methods using these search directions, with the predictor and corrector steps used either in series (similar to the Mizuno-Todd-Ye method) or in parallel (similar to Mizuno et al./McShane's method). The methods attain global linear convergence with a convergence ratio which, depending on the quality of the starting iterate, ranges from 1 ? O(p n) ?1 to 1 ? O(n) ?1. Our analysis is fairly simple and parallels that for the LP and LCP cases.

[1]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[2]  S. Mizuno,et al.  PRACTICAL POLYNOMIAL TIME ALGORITHMS FOR LINEAR COMPLEMENTARITY PROBLEMS , 1989 .

[3]  Paul Tseng,et al.  Complexity analysis of a linear complementarity algorithm based on a Lyapunov function , 1992, Math. Program..

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

[5]  Shinji Mizuno,et al.  A primal—dual infeasible-interior-point algorithm for linear programming , 1993, Math. Program..

[6]  Shinji Mizuno,et al.  On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming , 1993, Math. Oper. Res..

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

[8]  Y. Nesterov,et al.  Self-Scaled Cones and Interior-Point Methods in Nonlinear Programming , 1994 .

[9]  Shinji Mizuno,et al.  Polynomiality of infeasible-interior-point algorithms for linear programming , 1994, Math. Program..

[10]  P. TsengyJune.  Simpliied Analysis of an O(nl)-iteration Infeasible Predictor-corrector Path-following Method for Monotone Lcp , 1994 .

[11]  Kevin A. McShane.  Superlinearly Convergent O(√(n) L)-Iteration Interior-Point Algorithms for Linear Programming and the Monotone Linear Complementarity Problem , 1994, SIAM J. Optim..

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

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

[14]  F. Potra,et al.  Homogeneous Interior{point Algorithms for Semideenite Programming , 1995 .

[15]  Chih-Jen Lin,et al.  A predictor corrector method for semi-definite linear programming , 1995 .

[16]  R. Saigal,et al.  An Infeasible Start Predictor Corrector Method for Semi-deenite Linear Programming , 1995 .

[17]  Chih-Jen Lin,et al.  An infeasable start predictor corrector method for semi-definite linear programming , 1995 .

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

[19]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

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

[21]  Clóvis C. Gonzaga,et al.  The largest step path following algorithm for monotone linear complementarity problems , 1997, Math. Program..

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

[23]  Leonid Faybusovich,et al.  Semidefinite Programming: A Path-Following Algorithm for a Linear-Quadratic Functional , 1996, SIAM J. Optim..

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

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

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

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

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

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

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