A New Full Nesterov–Todd Step Primal–Dual Path-Following Interior-Point Algorithm for Symmetric Optimization

In this paper, we generalize a primal–dual path-following interior-point algorithm for linear optimization to symmetric optimization by using Euclidean Jordan algebras. The proposed algorithm is based on a new technique for finding the search directions and the strategy of the central path. At each iteration, we use only full Nesterov–Todd steps. Moreover, we derive the currently best known iteration bound for the small-update method. This unifies the analysis for linear, second-order cone, and semidefinite optimizations.

[1]  Bharath Kumar Rangarajan,et al.  Polynomial Convergence of Infeasible-Interior-Point Methods over Symmetric Cones , 2006, SIAM J. Optim..

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

[3]  Zsolt Darvay.  New Interior Point Algorithms in Linear Programming , 2003 .

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

[5]  Osman Güler,et al.  Barrier Functions in Interior Point Methods , 1996, Math. Oper. Res..

[6]  L. Faybusovich.  A Jordan-algebraic approach to potential-reduction algorithms , 2002 .

[7]  Farid Alizadeh,et al.  Extension of primal-dual interior point algorithms to symmetric cones , 2003, Math. Program..

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

[9]  E. D. Klerk,et al.  Aspects of semidefinite programming : interior point algorithms and selected applications , 2002 .

[10]  M. Muramatsu.  On a Commutative Class of Search Directions for Linear Programming over Symmetric Cones , 2002 .

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

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

[13]  Kees Roos,et al.  A Full-Newton Step O(n) Infeasible Interior-Point Algorithm for Linear Optimization , 2006, SIAM J. Optim..

[14]  Yan-Qin Bai,et al.  A primal-dual interior-point algorithm for second-order cone optimization with full Nesterov-Todd step , 2009, Appl. Math. Comput..

[15]  Guoyong Gu,et al.  Full Nesterov-Todd step infeasible interior-point method for symmetric optimization , 2011, Eur. J. Oper. Res..

[16]  Yan-Qin Bai,et al.  A new primal-dual path-following interior-point algorithm for semidefinite optimization , 2009 .

[17]  Z. Luo,et al.  Conic convex programming and self-dual embedding , 1998 .

[18]  Mohamed Achache,et al.  A new primal-dual path-following method for convex quadratic programming , 2006 .

[19]  R. Saigal,et al.  Handbook of semidefinite programming : theory, algorithms, and applications , 2000 .

[20]  Jun Ji,et al.  On the Local Convergence of a Predictor-Corrector Method for Semidefinite Programming , 1999, SIAM J. Optim..

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

[22]  F. Potra,et al.  On homogeneous interrior-point algorithms for semidefinite programming , 1998 .

[23]  Manuel V. C. Vieira,et al.  Jordan algebraic approach to symmetric optimization , 2007 .

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

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