Post-optimality sensitivity analysis in abstract spaces with applications to continuous-time programming problems

Abstract

The programming problem under consideration consists in maximizing a concave objective functional, subject to convex operator inequality contraints. The assumptions include the existence of an optimum solution, Frechet differentiability of all operators involved, and the existence of the topological complement of the null space of the Frechet derivative of the constraint operator. It is shown that the rate of change of the optimum value of the objective functional due to the perturbation is measured by the dual. The optimum values of the primal variables are locally approximated as linear functions of the perturbation; the theory of generalized inverse operators is used in the approximation. We give an approximation to the primal variables if the problem is perturbed. The results are specialized for some continuous-time and finite-dimensional cases. Two examples for finite-dimensional problems are given. We apply the theory to the continuous-time linear programming problem and prove some continuity results for the optimal primal and dual objective functionals.