Second order optimality conditions for minimization on a general set. Part 1: Applications to mathematical programming

This paper is devoted to second-order optimality conditions for minimization of a C2 function f on a general set K in a Banach space X. We consider both necessary and sufficient conditions of the second-order which differ by the strengthening of inequalities in their formulations. The conditions use first and second order approximations (first and second-order tangents) of the set K. The no gap sufficient conditions need additional assumptions in comparison with necessary conditions. We show that these assumptions hold true in the case when the set K is an intersection of a finite number of sets described by smooth inequalities and equalities, like in problems of the mathematical programming. Moreover, we illustrate the new conditions by deducing some mathematical programming results. In this sense the paper is partly a survey. One non-trivial illustrative example in an infinite dimensional space concerns the case when K can not be represented as an intersection described above. The novelty of our approach is due, on one hand, to the arbitrariness of the set K, and on the other hand, to quite straightforward proofs.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates second-order optimality conditions for the minimization problem of a C2 function f on a general set K in a Banach space X. Both necessary and sufficient conditions are discussed, with the sufficiency condition requiring additional assumptions. The paper demonstrates the validity of these assumptions for the case when the set K is an intersection of sets described by smooth inequalities and equalities, such as in mathematical programming problems. The novelty of the approach lies in the arbitrary nature of the set K and the straightforward proofs.