Generalized Sensitivity Analysis of Nonlinear Programs

This paper extends classical sensitivity results for nonlinear programs to cases in which parametric perturbations cause changes in the active set. This is accomplished using lexicographic directional derivatives, a recently developed tool in nonsmooth analysis based on Nesterov's lexicographic differentiation. A nonsmooth implicit function theorem is augmented with generalized derivative information and applied to a standard nonsmooth reformulation of the parametric KKT system. It is shown that the sufficient conditions for this implicit function theorem variant are implied by a KKT point satisfying the linear independence constraint qualification and strong second-order sufficiency. Mirroring the classical theory, the resulting sensitivity system is a nonsmooth equation system which admits primal and dual sensitivities as its unique solution. Practically implementable algorithms are provided for calculating the nonsmooth sensitivity system's unique solution, which is then used to furnish B-subdifferential elements of the primal and dual variable solutions by solving a linear equation system. Consequently, the findings in this article are computationally relevant since dedicated nonsmooth equation-solving and optimization methods display attractive convergence properties when supplied with such generalized derivative elements. The results have potential applications in nonlinear model predictive control and problems involving dynamic systems with mathematical programs embedded. Extending the theoretical treatments here to sensitivity analysis theory of other mathematical programs is also anticipated.