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.

An Active Set Algorithm for Nonlinear Optimization with Polyhedral Constraints

A polyhedral active set algorithm PASA is developed for solving a nonlinear optimization problem whose feasible set is a polyhedron. Phase one of the algorithm is the gradient projection method, while phase two is any algorithm for solving a linearly constrained optimization problem. Rules are provided for branching between the two phases. Global convergence to a stationary point is established, while asymptotically PASA performs only phase two when either a nondegeneracy assumption holds, or the active constraints are linearly independent and a strong second-order sufficient optimality condition holds

Stationary Point Sets: Convex Quadratic Optimization is Universal in Nonlinear Optimization

We investigate the local topological structure, stationary point sets in parametric optimization genericly may have. Our main result states that, up to stratified isomorphism, any such structure is already present in the small subclass of parametric problems with convex quadratic objective function and affine-linear constraints. In other words, the convex quadratic problems produce a normal form for the local topological structure of stationary point sets. As a consequence we see, as far as no equality constraints are involved, that the closure of the stationary point set constitutes a manifold with boundary. The boundary is exactly the violation set of the Mangasarian Fromovitz constraint qualification. A side result states that stationary point sets and violation sets of Mangasarian Fromovitz constraint qualification carry the same set of possible local structures as stratified spaces

An axiomatization of the Euclidean compromise solution

The utopia point of a multicriteria optimization problem is the vector that specifies for each criterion the most favourable among the feasible values. The Euclidean compromise solution in multicriteria optimization is a solution concept that assigns to a feasible set the alternative with minimal Euclidean distance to the utopia point. The purpose of this paper is to provide a characterization of the Euclidean compromise solution. Consistency plays a crucial role in our approach.Consistency; Euclidean compromise solution; Multicriteria optimization