Full-Newton step infeasible interior-point algorithm for SDO problems

In this paper we propose a primal-dual path-following interior-point algorithm for semidefinite optimization. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. Each main step of the algorithm consists of a feasibility step and several centering steps. At each iteration, we use only full-Newton step. Moreover, we use a more natural feasibility step, which targets at the $\mu^+$-center. The iteration bound of the algorithm coincides with the currently best iteration bound for semidefinite optimization problems.

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

[2]  Hossein Mansouri,et al.  A Path-Following Infeasible Interior-Point Algorithm for Semidefinite Programming , 2012 .

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

[4]  Irvin Lustig,et al.  Feasibility issues in a primal-dual interior-point method for linear programming , 1990, Math. Program..

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

[6]  Z. Luo,et al.  Superlinear Convergence of a Symmetric Primal-Dual Path Following Algorithm for SDP , 1998 .

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

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

[9]  Hossein Mansouri,et al.  A new full-Newton step O(n) infeasible interior-point algorithm for semidefinite optimization , 2009, Numerical Algorithms.

[10]  H. Mansouri.  Full-Newton step interior-point methods for conic optimization , 2008 .

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

[12]  H. Luetkepohl.  The Handbook of Matrices , 1996 .

[13]  C. Helmberg.  Semidefinite Programming for Combinatorial Optimization , 2000 .

[14]  Nimrod Megiddo,et al.  A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems , 1991, Lecture Notes in Computer Science.

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

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

[17]  Jiming Peng,et al.  Self-regular functions and new search directions for linear and semidefinite optimization , 2002, Math. Program..

[18]  Etienne de Klerk,et al.  Aspects of Semidefinite Programming , 2002 .

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