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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)$\le 0\}$, X is an open subset of ${\bbfR}\sp n$ and $f: X\to {\bbfR}$, $g: X\to {\bbfR}\sp m$ 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 $x\sb 0\in P$ is optimal for (1) if and only if the Kuhn-Tucker conditions hold at $x\sb 0$ and there exists a function $\eta$ : $P\to {\bbfR}\sp n$, $\eta\ne 0$, such that $f(x)-f(x\sb 0)\ge [\nabla\sb xf(x\sb 0)]\sp T\eta (x)$ and $-g(x\sb 0)\ge [\nabla\sb xg(x\sb 0)]\sp T\eta (x)$ for all $x\in P$. Necessary and sufficient conditions are also given for optimality of the problem dual to (1).
Reviewer: M.Studniarski

49K10Free problems in several independent variables (optimality conditions)
49N15Duality theory (optimization)
90C30Nonlinear programming
Full Text: DOI
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