A new barrier for a class of semidefinite problems

Issue		RAIRO Oper. Res. Volume 40, Number 3, July-September 2006


Page(s)		303 - 323
DOI		10.1051/ro:2006022
Published online		08 November 2006

RAIRO Oper. Res. 40 (2006) 303-323
DOI: 10.1051/ro:2006022

A new barrier for a class of semidefinite problems

Erik A. Papa Quiroz¹ and Paolo Roberto Oliveira^{1, 2}

¹ Programa de Engenharia de Sistemas e Computacão, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil; erik@cos.ufrj.br
² Rua Honorio, 1144 c 5, 20771-421, Rio de Janeiro, Brazil; poliveir@cos.ufrj.br

(Received April 4, 2005. Accepted March 28, 2006. Published online 8 November 2006.)

Abstract
We introduce a new barrier function to solve a class of Semidefinite Optimization Problems (SOP) with bounded variables. That class is motivated by some (SOP) as the minimization of the sum of the first few eigenvalues of symmetric matrices and graph partitioning problems. We study the primal-dual central path defined by the new barrier and we show that this path is analytic, bounded and that all cluster points are optimal solutions of the primal-dual pair of problems. Then, using some ideas from semi-analytic geometry we prove its full convergence. Finally, we introduce a new proximal point algorithm for that class of problems and prove its convergence.

Key words: Interior point methods, barrier function, central path, semidefinite optimization.

© EDP Sciences 2006

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