Strong Duality and Solution Existence Under Minimal Assumptions in Conic Linear Programming

Conic linear programs in locally convex Hausdorff topological vector spaces are addressed in this paper. Solution existence for the dual problem, as well as solution existence for the primal problem, and strong duality, are proved under minimal regularity assumptions. Namely, to get the results and a Farkas-type theorem for infinite-dimensional conic linear inequalities, we employ the generalized Slater condition either for the primal problem or for the dual problem, as well as proper separation and the concept of quasi-regularity of convex sets. Illustrative examples are presented.