A Potential Reduction Newton Method for Constrained Equations

Extending our previous work [T. Wang, R. D. C. Monteiro, and J.-S. Pang, Math. Programming, 74 (1996), pp. 159--195], this paper presents a general potential reduction Newton method for solving a constrained system of nonlinear equations. A major convergence result for the method is established. Specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed. Strengthened convergence results are established in the context of these specializations.

[1]  P. Tseng.  Search Directions And Convergence Analysis Of Some Infeasible Path-Following Methods For The Monoton , 1996 .

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

[3]  Renato D. C. Monteiro,et al.  On Two Interior-Point Mappings for Nonlinear Semidefinite Complementarity Problems , 1998, Math. Oper. Res..

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

[5]  Renato D. C. Monteiro,et al.  A note on the existence of the Alizadeh-Haeberly-Overton direction for semidefinite programming , 1997, Math. Program..

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

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

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

[9]  Takashi Tsuchiya,et al.  Polynomial Convergence of a New Family of Primal-Dual Algorithms for Semidefinite Programming , 1999, SIAM J. Optim..

[10]  Tao Wang,et al.  An interior point potential reduction method for constrained equations , 1996, Math. Program..

[11]  Nimrod Megiddo,et al.  A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems , 1991, Lecture Notes in Computer Science.

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

[13]  Alexander Shapiro,et al.  First and second order analysis of nonlinear semidefinite programs , 1997, Math. Program..

[14]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[15]  R. D. Murphy,et al.  Iterative solution of nonlinear equations , 1994 .

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

[17]  P. Tseng.  Search directions and convergence analysis of some infeasibnle path-following methods for the monoton semi-definite lcp ∗ , 1998 .

[18]  N. Megiddo.  Pathways to the optimal set in linear programming , 1989 .

[19]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

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

[21]  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..

[22]  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..

[23]  Renato D. C. Monteiro,et al.  Properties of an Interior-Point Mapping for Mixed Complementarity Problems , 1996, Math. Oper. Res..

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

[25]  Takashi Tsuchiya,et al.  Polynomiality of primal-dual algorithms for semidefinite linear complementarity problems based on the Kojima-Shindoh-Hara family of directions , 1999, Math. Program..

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

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

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

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

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

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

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

[33]  A. Shapiro.  First and Second Order Analysis of Nonlinear Semideenite Programs , 1997 .

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