Linear conic and two-stage stochastic optimization revisited via semi-infinite optimization

In this paper, we update the theory of deterministic linear semi-infinite programming, mainly with the dual characterizations of the constraint qualifications, which play a crucial role in optimality and duality. From this theory, we obtain new results on conic and two-stage stochastic linear optimization. Specifically, for conic linear optimization problems, we characterize the existence of feasible solutions and some geometric properties of the feasible set, and we also provide theorems on optimality and duality. Analogously, regarding stochastic optimization problems, we study the semi-infinite reformulation of a problem-based scenario reduction problem in two-stage stochastic linear programming, providing a sufficient condition for the existence of feasible solutions as well as optimality and duality theorems to its non-combinatorial part.