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

The authors consider the following nonlinear programming problem: \[ (1)\quad \min imize\quad f(x)\quad subject\quad to\quad x\in P, \] where \(P=\{x\in X:\) q(x)\(\leq 0\}\), X is an open subset of \({\mathbb{R}}^ n\) and \(f: X\to {\mathbb{R}}\), \(g: X\to {\mathbb{R}}^ m\) 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 \(x_ 0\in P\) is optimal for (1) if and only if the Kuhn-Tucker conditions hold at \(x_ 0\) and there exists a function \(\eta\) : \(P\to {\mathbb{R}}^ n\), \(\eta\neq 0\), such that \(f(x)-f(x_ 0)\geq [\nabla_ xf(x_ 0)]^ T\eta (x)\) and \(-g(x_ 0)\geq [\nabla_ xg(x_ 0)]^ T\eta (x)\) for all \(x\in P\). Necessary and sufficient conditions are also given for optimality of the problem dual to (1).
Reviewer: M.Studniarski

MSC:

49K10 Optimality conditions for free problems in two or more independent variables
49N15 Duality theory (optimization)
90C30 Nonlinear programming
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References:

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