On the convergence of the entropy-exponential penalty trajectories and generalized proximal point methods in semidefinite optimization

Authors:
O. P. Ferreira;P. R. Oliveira;R. C. Silva
Affiliations:
IME, Universidade Federal de Goiás, Goiania, Brazil 74001-970;COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil 21945-970;DM/ICE, Universidade Federal do Amazonas, Manaus, Brazil 69077-000
Venue:
Journal of Global Optimization
Year:
2009

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Abstract

The convergence of primal and dual central paths associated to entropy and exponential functions, respectively, for semidefinite programming problem are studied in this paper. It is proved that the primal path converges to the analytic center of the primal optimal set with respect to the entropy function, the dual path converges to a point in the dual optimal set and the primal-dual path associated to this paths converges to a point in the primal-dual optimal set. As an application, the generalized proximal point method with the Kullback-Leibler distance applied to semidefinite programming problems is considered. The convergence of the primal proximal sequence to the analytic center of the primal optimal set with respect to the entropy function is established and the convergence of a particular weighted dual proximal sequence to a point in the dual optimal set is obtained.