Variational approximations of a dual pair of mathematical programming problems
Huynh Thi Hong Diem · Mathematical Analysis and Convex Optimization · 2020
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). MSC(2010): 90C30, 90C31, 90C47, 49J53