The generalized jacobian of the optimal solution in parametric optimization

Abstract

Abstract If a strong sufficient optimality condition of second order together with the Mangasarian-Fromowitz and the constant rank constraint qualifications are: satisfied for a parametric optimization problem, then a local optimal solution is strongly stable in the sense of Kojima and the corresponding optimal solution function is locally Lipschitz continuous. In the article the possibilities for the computation of the generalized Jacobian of this function are discussed. We will give formulae for the guaranteed computation of the entire generalized Jacobian, provided that an additional assumption is satisfied. An example will show its necessity. Unfortunately, this assumption is difficult to be verified. Without it, at least one element of the generalized Jacobian can be computed with non-polynomial complexity in the worst case. Using a uniforrn distribution in the parameter space, a last approach yields one of them with probability one in polynomial time Keywords: Nonlinear Parametric OptimizationLipschitz ContinuityGeneralized Differentiability PC 1-Functions ∗Corresponding author ∗Corresponding author Notes ∗Corresponding author