Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions

Abstract.In this paper we study primal-dual path-following algorithms for the second-order cone programming (SOCP) based on a family of directions that is a natural extension of the Monteiro-Zhang (MZ) family for semidefinite programming. We show that the polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima et al. and Monteiro and Adler, and the predictor-corrector algorithm of Mizuno et al., carry over to the context of SOCP, that is they have an O( logε-1) iteration-complexity to reduce the duality gap by a factor of ε, where n is the number of second-order cones. Since the MZ-type family studied in this paper includes an analogue of the Alizadeh, Haeberly and Overton pure Newton direction, we establish for the first time the polynomial convergence of primal-dual algorithms for SOCP based on this search direction.

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

[2]  Shinji Mizuno,et al.  A polynomial-time algorithm for a class of linear complementarity problems , 1989, Math. Program..

[3]  Renato D. C. Monteiro,et al.  Interior path following primal-dual algorithms. part II: Convex quadratic programming , 1989, Math. Program..

[4]  Renato D. C. Monteiro,et al.  Interior path following primal-dual algorithms. part I: Linear programming , 1989, Math. Program..

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

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

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

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

[9]  J. Faraut,et al.  Analysis on Symmetric Cones , 1995 .

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

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

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

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

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

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

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

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

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

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

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

[21]  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..

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

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

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

[25]  T. Tsuchiya.  A Convergence Analysis of the Scaling-invariant Primal-dual Path-following Algorithms for Second-ord , 1998 .

[26]  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..

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

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

[29]  J. Sturm.  Similarity and other spectral relations for symmetric cones , 2000 .