On Extending Some Primal-Dual Interior-Point Algorithms From Linear Programming to Semidefinite Programming

This work concerns primal--dual interior-point methods for semidefinite programming (SDP) that use a search direction originally proposed by Helmberg et al. [SIAM J. Optim., 6 (1996), pp. 342--361] and Kojima, Shindoh, and Hara [SIAM J. Optim., 7 (1997), pp. 86--125.] and recently rediscovered by Monteiro [SIAM J. Optim., 7 (1997), pp. 663--678] in a more explicit form. In analyzing these methods, a number of basic equalities and inequalities were developed in [Kojima, Shindoh, and Hara] and also in [Monteiro] through different means and in different forms. In this paper, we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming (LP), producing basic relationships that have compact forms almost identical to their counterparts in LP. We also introduce a new formulation of the central path and variable-metric measures of centrality. These results provide convenient tools for deriving polynomiality results for primal--dual algorithms extended from LP to SDP using the aforementioned and related search directions. We present examples of such extensions, including the long-step infeasible-interior-point algorithm of Zhang [SIAM J. Optim., 4 (1994), pp. 208--227].

[1]  M. Overton.  On minimizing the maximum eigenvalue of a symmetric matrix , 1988 .

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

[3]  Sanjay Mehrotra,et al.  On the Implementation of a Primal-Dual Interior Point Method , 1992, SIAM J. Optim..

[4]  Michael L. Overton,et al.  Large-Scale Optimization of Eigenvalues , 1990, SIAM J. Optim..

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

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

[7]  R. Vanderbei,et al.  Interior point methods for max - min eigenvalue problems , 1993 .

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

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

[10]  Yin Zhang,et al.  On the Convergence of a Class of Infeasible Interior-Point Methods for the Horizontal Linear Complementarity Problem , 1994, SIAM J. Optim..

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

[12]  Yin Zhang,et al.  On polynomiality of the Mehrotra-type predictor—corrector interior-point algorithms , 1995, Math. Program..

[13]  Shinji Hara,et al.  Interior Point Methods for the Monotone Linear Complementarity Problem in Symmetric Matrices , 1995 .

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

[15]  Motakuri V. Ramana,et al.  An exact duality theory for semidefinite programming and its complexity implications , 1997, Math. Program..

[16]  M. Kojima,et al.  REDUCTION OF MONOTONE LINEAR COMPLEMENTARITY PROBLEMS OVER CONES TO LINEAR PROGRAMS OVER CONES , 1997 .

[17]  Masakazu Kojima,et al.  Exploiting sparsity in primal-dual interior-point methods for semidefinite programming , 1997, Math. Program..

[18]  Michael L. Overton,et al.  Complementarity and nondegeneracy in semidefinite programming , 1997, Math. Program..

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

[20]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

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