Conditions for Optimality and Strong Stability in Nonlinear Programs without assuming Twice Differentiability of Data

Klatte, D., Kummer, B., & Walzebok, R. (1989). Conditions for Optimality and Strong Stability in Nonlinear Programs without assuming Twice Differentiability of Data. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-89-089

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Abstract

The present paper is concerned with optimization problems in which the data are differentiable functions having a continuous or locally Lipschitzian gradient mapping. Its main purpose is to develop second-order sufficient conditions for a stationary solution to a program with C^{1,1} data to be a strict local minimizer or to be a local minimizer which is even strongly stable with respect to certain perturbations of the data. It turns out that some concept of a set-valued directional derivative of a Lipschitzian mapping is a suitable tool to extend well-known results in the case of programs with twice differentiable data to more general situations. The local minimizers being under consideration have to satisfy the Mangasarian-Fromovitz CQ. An application to iterated local minimization is sketched.

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