Primal-Dual Path-Following Algorithms for Semidefinite Programming

This paper deals with a class of primal--dual interior-point algorithms for semidefinite programming (SDP) which was recently introduced by Kojima, Shindoh, and Hara [SIAM J. Optim., 7 (1997), pp. 86--125]. These authors proposed a family of primal-dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix X1/2S-1/2. They study three primal--dual algorithms based on this family of search directions: a short-step path-following method, a feasible potential-reduction method, and an infeasible potential-reduction method. However, they were not able to provide an algorithm which generalizes the long-step path-following algorithm introduced by Kojima, Mizuno, and Yoshise [Progress in Mathematical Programming: Interior Point and Related Methods, N. Megiddor, ed., Springer-Verlag, Berlin, New York, 1989, pp. 29--47]. In this paper, we characterize two search directions within their family as being (unique) solutions of systems of linear equations in symmetric variables. Based on this characterization, we present a simplified polynomial convergence proof for one of their short-step path-following algorithms and, for the first time, a polynomially convergent long-step path-following algorithm for SDP which requires an extra $\sqrt{n}$ factor in its iteration-complexity order as compared to its linear programming counterpart, where n is the number of rows (or columns) of the matrices involved.

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

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

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

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

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

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

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

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

[9]  Roy E. Marsten,et al.  Feature Article - Interior Point Methods for Linear Programming: Computational State of the Art , 1994, INFORMS J. Comput..

[10]  B. Jansen,et al.  Primal-dual algorithms for linear programming based on the logarithmic barrier method , 1994 .

[11]  Leonid Faybusovich.  On a Matrix Generalization of Affine-Scaling Vector Fields , 1995, SIAM J. Matrix Anal. Appl..

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

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

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

[15]  B. He,et al.  Method of approximate centers for semi-definite programming , 1997 .

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

[17]  Etienne de Klerk,et al.  Initialization in semidefinite programming via a self-dual skew-symmetric embedding , 1997, Oper. Res. Lett..

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

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

[20]  Jean-Philippe Vial,et al.  Theory and algorithms for linear optimization - an interior point approach , 1998, Wiley-Interscience series in discrete mathematics and optimization.

[21]  Etienne de Klerk,et al.  Polynomial Primal-Dual Affine Scaling Algorithms in Semidefinite Programming , 1998, J. Comb. Optim..

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

[23]  Jianmin Jiang,et al.  A long-step primal-dual path-following method for semidefinite programming , 1998, Oper. Res. Lett..

[24]  Katya Scheinberg,et al.  Interior Point Trajectories in Semidefinite Programming , 1998, SIAM J. Optim..

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