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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