Sensitivity analysis for abnormal optimization problems with a cone constraint
The authors analyze the sensitivity of optimal values and optimal sets of finite dimensional optimization problems depending on parameters. par The feasible set is defined by means of a constraint mapping F and a fixed closed convex cone K. Both the objective function f and the constraint mapping F are smooth and depend on a parameter. The authors do not suppose the classical regularity assumption of S. M. Robinson (nor the weaker directional regularity). The analysis in this paper is based on the concept of (directional) 2-regularity for the constraint mapping F with respect to the cone K. Roughly speaking, the 2-regularity uses second order information of F in order to fill in a surjectivity gap with respect to the linearization of F (which then makes an approach via implicit functions possible). par The authors derive estimates both from above and below for the optimal value function (with Hölder exponent equal to 0.5). Moreover, an analogous Hölder estimate on the distance of the optimal set of the perturbed problem with respect to the optimal point of the unperturbed problem is presented.