Unicity Results for General Linear Semi-Infinite Optimization Problems Using a New Concept of Active Constraints

We consider parametric semi-infinite optimization problems without the usual assumptions on the continuity of the involved mappings and on the compactness of the index set counting the inequalities. We establish a characterization of those optimization problems which have a unique or strongly unique solution and which are stable under small perturbations. This result generalizes a well-known theorem of Nuernberger. The crucial roles in our investigations are a new concept of active constraints, a generalized Slater's condition, and a Kuhn-Tucker-type theorem. Finally, we give some applications in vector optimization, for approximation problems in normed spaces, and in the stability of the minimal value