Páles, Zs.; Zeidan, V. M. Nonsmooth optimum problems with constraints. (English) Zbl 0821.49020 SIAM J. Control Optimization 32, No. 5, 1476-1502 (1994). In this paper, the authors develop second-order necessary optimality conditions for nonsmooth infinite-dimensional optimization problems with Banach space-valued equality and real-valued inequality constraints. Another constraint of type \(z \in Q\) is present, where \(Q\) is a closed convex subset of a Banach space \(Z\). The objective function is \(F(z) = \sup_{t \in T} f(t,z)\), \(T\) being a compact metric space. The paper also includes a unified presentation of the notions and results introduced by A. Ya. Dubovitskij and A. A. Milyutin [Dokl. Akad. Nauk SSSR 149, 759-762 (1963; Zbl 0133.055)]. The authors define the concepts of descent, admissible, and tangent variations or order \(k\) for a one-parameter family of directions. The suitable choice of the family of directions leads to the first-, second-, and higher-order necessary conditions. A detailed study of the second order variations leads to the main result of the paper: a second-order Lagrange multiplier rule. Reviewer: E.Casas (Santander) Cited in 32 Documents MSC: 49K27 Optimality conditions for problems in abstract spaces 49M20 Numerical methods of relaxation type 49J52 Nonsmooth analysis Keywords:nonsmooth analysis; Dubovitskij-Milyutin approach; envelope-like effect; second-order necessary optimality conditions; inequality constraints Citations:Zbl 0133.055 PDFBibTeX XMLCite \textit{Zs. Páles} and \textit{V. M. Zeidan}, SIAM J. Control Optim. 32, No. 5, 1476--1502 (1994; Zbl 0821.49020) Full Text: DOI