On a commutative class of search directions for linear programming over symmetric cones

Authors:
M. Muramatsu
Affiliations:
Associate Professor, Department of Computer Science, The University of Electro-Communications, Chofu-Shi, Tokyo, Japan
Venue:
Journal of Optimization Theory and Applications
Year:
2002

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Abstract

The commutative class of search directions for semidefinite programming was first proposed by Monteiro and Zhang (Ref. 1). In this paper, we investigate the corresponding class of search directions for linear programming over symmetric cones, which is a class of convex optimization problems including linear programming, second-order cone programming, and semidefinite progrmming as special cases. Complexity results are established for short-step, semilong-step, and long-step algorithms. Then, we propose a subclass of the commutative class for which we can prove polynomial complexities of the interior-point method using semilong steps and long steps. This subclass still contains the Nesterov-Todd direction and the Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro direction. An explicit formula to calculate any member of the class is also given.