variational approximations of a dual pair of mathematical programming problems

We study variational approximations of a dual pair of mathematical programming problems in terms of epi/hypo-convergence and inside epi/hypo-convergence of approximating lagrange functions of the pair. first, the painlevé-kuratowski convergence of approximate saddle points of approximating lagrange functions is established under the inside epi/hypo-convergence of these approximating lagrange functions or under types of inside convergence directly of the data of problems. from this, we obtain a couple of solutions of the pair of problems and a strong duality. under a stronger variational convergence called ancillary tight epi/hypo-convergence, we obtain the painlevé-kuratowski convergence of approximate minsup-points and approximate maxinf-points of approximating lagrange functions (when approximate saddle points are not necessary to exist).