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Necessary and sufficient conditions in constrained optimization. (English) Zbl 0622.49005

The authors consider the following nonlinear programming problem:

(1)minimizef(x)subjecttoxP,

where P={xX: q(x)0}, X is an open subset of n and f:X, g:X m are differentiable functions. The aim of the paper is to give a set of conditions which are both necessary and sufficient for optimality in problem (1). It is shown (under some additional assumptions) that x 0 P is optimal for (1) if and only if the Kuhn-Tucker conditions hold at x 0 and there exists a function η : P n , η0, such that f(x)-f(x 0 )[ x f(x 0 )] T η(x) and -g(x 0 )[ x g(x 0 )] T η(x) for all xP. Necessary and sufficient conditions are also given for optimality of the problem dual to (1).

Reviewer: M.Studniarski
MSC:
49K10Free problems in several independent variables (optimality conditions)
49N15Duality theory (optimization)
90C30Nonlinear programming
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