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Nonsmooth optimum problems with constraints. (English) Zbl 0821.49020

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.

MSC:

49K27 Optimality conditions for problems in abstract spaces
49M20 Numerical methods of relaxation type
49J52 Nonsmooth analysis

Citations:

Zbl 0133.055
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