On sensitivity of central solutions in semidefinite programming

Citation
Jf. Sturm et S. Zhang, On sensitivity of central solutions in semidefinite programming, MATH PROGR, 90(2), 2001, pp. 205-227
Citations number
38
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Mathematics
Journal title
MATHEMATICAL PROGRAMMING
ISSN journal
0025-5610 → ACNP
Volume
90
Issue
2
Year of publication
2001
Pages
205 - 227
Database
ISI
SICI code
0025-5610(200104)90:2<205:OSOCSI>2.0.ZU;2-H
Abstract
In this paper we study the properties of the analytic central path of a sem idefinite programming problem under perturbation of the right hand side of the constraints, including the limiting behavior when the central optimal s olution, namely the analytic center of the optimal set, is approached. Our analysis assumes the primal-dual Slater condition and the strict complement arity condition. Our findings are as follows. First, on the negative side, if we view the central optimal solution as a function of the right hand sid e of the constraints, then this function is not continuous in general, wher eas in the linear programming case this function is known to be Lipschitz c ontinuous. On the positive side, compared with the previous conclusion we o btain a (seemingly) paradoxical result: on the central path any directional derivative with respect to the right hand side of the constraints is bound ed, and even converges as the central optimal solution is approached. This phenomenon is possible due to the lack of a uniform bound on the derivative s with respect to the right hand side parameters. All these results are bas ed on the strict complementarity assumption. Concerning this last property we give an example. in that example the yet of right hand side parameters f or which the strict complementarity condition holds is neither open nor clo sed. This is remarkable since a similar set for which the primal-dual Slate r condition holds is always open.