Lipschitz continuity of the optimal value via bounds on the optimal set in linear semi-infinite optimization

We consider the parametric space of all the linear semi–infinite programming problems with constraint systems having the same index set. Under a certain regularity condition, the so–called well–posedness with respect to the solvability, it is known from Cánovas et al. that the optimal value function is Lipschitz continuous around the nominal problem π. In this paper we obtain an explicit Lipschitz constant for such a function in a certain neighborhood of π. We emphasize the fact that both the constant and the size of the neighborhood are exclusively expressed in terms of the nominal problem data, and that they involve the distances to primal and to dual inconsistency. Moreover, a uniform bound for the optimal set is provided. This bound constitutes a key ingredient to derive the Lipschitz constant for the optimal value function.