Error bounds for infinite systems of convex inequalities without Slater's condition

Authors
Citation
M. Gugat, Error bounds for infinite systems of convex inequalities without Slater's condition, MATH PROGR, 88(2), 2000, pp. 255-275
Citations number
23
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Mathematics
Journal title
MATHEMATICAL PROGRAMMING
ISSN journal
0025-5610 → ACNP
Volume
88
Issue
2
Year of publication
2000
Pages
255 - 275
Database
ISI
SICI code
0025-5610(200008)88:2<255:EBFISO>2.0.ZU;2-M
Abstract
The feasible set of a convex semi-infinite program is described by a possib ly infinite system of convex inequality constraints. We want to obtain an u pper bound for the distance of a given point from this set in terms of a co nstant multiplied by the value of the maximally violated constraint functio n in this point. Apart from this Lipschitz case we also consider error boun ds of Holder type, where the value of the residual of the constraints is ra ised to a certain power. We give sufficient conditions for the validity of such bounds, Our conditio ns do not require that the Slater condition is valid. For the definition of our conditions, we consider the projections on enlarged sets corresponding to relaxed constraints. We present a condition in terms of projection mult ipliers. a condition in terms of Slater points and a condition in terms of descent directions. For the Lipschitz case, we give five equivalent charact erizations of the validity of a global error bound. We extend previous results in two directions: First, we consider infinite s ystems of inequalities instead of finite systems, The second point is that we do not assume that the Slater condition holds which has been required in almost all earlier papers.