Sensitivity Analysis of a Stationary Point Set Map under Total Perturbations. Part 2: Robinson Stability

In Part 1 of this paper, we have estimated the Fr\'echet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint under total perturbations. From these estimates, necessary and sufficient conditions for the local Lipschitz-like property of the map have been obtained. In this part, we establish sufficient conditions for the Robinson stability of the stationary point set map. This allows us to revisit and extend several stability theorems in indefinite quadratic programming. A comparison of our results with the ones which can be obtained via another approach is also given.Comment: This manuscript is based on the paper "Sensitivity Analysis of a Stationary Point Set Map under Total Perturbations. Part 2: Robinson Stability" which has been pubplished in Journal of Optimization Theory and Applications (DOI: 10.1007/s10957-018-1295-4). We have added the Section 6 "Appendices" to the paper. This section presents two proofs of Lemmas 5.1 and 5.

Sensitivity Analysis of a Stationary Point Set Map under Total Perturbations. Part 1: Lipschitzian Stability

By applying some theorems of Levy and Mordukhovich (Math Program 99: 311--327, 2004) and other related results, we estimate the Fr\'echet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint under total perturbations. From the obtained formulas we derive necessary and sufficient conditions for the local Lipschitz-like property of the stationary point set map. This leads us to new insights into the preceding deep investigations of Levy and Mordukhovich in the above-cited paper and of Qui (J Optim Theory Appl 161: 398--429, 2014; J Glob Optim 65: 615--635, 2016).Comment: This paper has been published in Journal of Optimization Theory and Application