On stable uniqueness in linear semi-infinite optimization

This paper is intended to provide conditions for the stability of the strong uniqueness of the optimal solution of a given linear semi-in…nite optimization (LSIO) problem, in the sense of the maintaining of the strong uniqueness property under su¢ ciently small perturbations of all the data. We consider LSIO problems such that the family of gradients of all the constraints is unbounded, extending earlier results of Nürnberger for continuous LSIO problems, and of Helbig and Todorov for LSIO problems with bounded set of gradients. To do this we characterize the absolutely (a¢ nely) stable problems, i.e., those LSIO problems whose feasible set (its a¢ ne hull, respectively) remains constant under su¢ ciently small perturbations. Key words. Linear semi-in…nite optimization, stable strong uniqueness, extended Nürnberger condition, a¢ ne stability, absolute stability. 2000 Mathematics Subject Classi…cation. 90C05, 90C34, 15A39.