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

• Published in: Journal of mathematical analysis and applications Vol. 529; no. 2; p. 127384
• Main Authors: Frankowska, Hélène, Osmolovskii, Nikolai P.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.01.2024
• Subjects: Equality and inequality constraints; First and second order tangents; Lagrange function; Local minimum; Second order optimality conditions; Separation theorem; Local minimum; Equality and inequality constraints; Separation theorem; Second order optimality conditions; Lagrange function; First and second order tangents
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2023.127384

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.