Sensitivity analysis of parametrized programs via generalized equations

This paper investigates local behavior of optimal solutions of parameterized optimization problems with cone constraints in Banach spaces. The corresponding first-order optimality conditions are formulated in a form of generalized equations (variational inequalities) and solutions of these generalized equations are studied. It is shown that under certain second-order sufficient optimality conditions and a regularity assumption related to the associated Lagrange multipliers, the considered optimal solutions are Lipschitzian stable. This is compared with a similar result in Shapiro and Bonnans. Under the additional assumption of uniqueness of the Lagrange multipliers, first-order expansions of the optimal solutions are given in terms of solutions of auxiliary optimization problems. Finally, as an example, semi-infinite programming problems are discussed.