A Convergence Analysis of the Scaling-invariant Primal-dual Path-following Algorithms for Second-ord

In this paper we study primal-dual path-following algorithms for second-order cone programming problems (SOCP). We extend the standard long-step/semilong-step/short-step primal-dual path-following alogorithms for LP and SDP to SOCP, and prove that the long-step algorithm using the NT direction and the HRVW/KSH/M direction have O(n log e-1 ) iteration-complexity and O(n3/2log e-1 ) iteration-complexity, respectively, to reduce the duality gap by a factor of 1/e, where n is the number of the second-order cones. We also show that the short and semilong-step algorithms using the NT direction and the HRVW/KSH/M direction have and O(n log e-1 ) iteration-complexities, respectively.

[1]  M. Kojima,et al.  A primal-dual interior point algorithm for linear programming , 1988 .

[2]  Florian Jarre,et al.  On the convergence of the method of analytic centers when applied to convex quadratic programs , 1991, Math. Program..

[3]  Sanjay Mehrotra,et al.  A method of analytic centers for quadratically constrained convex quadratic programs , 1991 .

[4]  Arkadi Nemirovski,et al.  Potential Reduction Polynomial Time Method for Truss Topology Design , 1994, SIAM J. Optim..

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

[6]  Katya Scheinberg,et al.  Extension of Karmarkar's algorithm onto convex quadratically constrained quadratic problems , 1996, Math. Program..

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

[8]  L. Faybusovich.  Linear systems in Jordan algebras and primal-dual interior-point algorithms , 1997 .

[9]  F. Alizadeh,et al.  Optimization with Semidefinite, Quadratic and Linear Constraints , 1997 .

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

[11]  L. Faybusovich.  Euclidean Jordan Algebras and Interior-point Algorithms , 1997 .

[12]  Yinyu Ye,et al.  An Efficient Algorithm for Minimizing a Sum of Euclidean Norms with Applications , 1997, SIAM J. Optim..

[13]  T. Tsuchiya.  A Polynomial Primal-Dual Path-Following Algorithm for Second-order Cone Programming , 1997 .

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

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

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

[17]  Florian Jarre,et al.  Optimal Truss Design by Interior-Point Methods , 1998, SIAM J. Optim..

[18]  F. Alizadeh,et al.  Associative Algebras, Symmetric Cones and Polynomial Time Interior Point Algorithms , 1998 .

[19]  Yin Zhang,et al.  A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming , 1998, Math. Program..

[20]  Stephen P. Boyd,et al.  Applications of second-order cone programming , 1998 .

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

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

[23]  Takashi Tsuchiya,et al.  Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions , 2000, Math. Program..