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

The authors consider the following nonlinear programming problem:

$\left(1\right)\phantom{\rule{1.em}{0ex}}minimize\phantom{\rule{1.em}{0ex}}f\left(x\right)\phantom{\rule{1.em}{0ex}}subject\phantom{\rule{1.em}{0ex}}to\phantom{\rule{1.em}{0ex}}x\in P,$

where $P=\left\{x\in X:$ q(x)$\le 0\right\}$, X is an open subset of ${ℝ}^{n}$ and $f:X\to ℝ$, $g:X\to {ℝ}^{m}$ 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 $\eta$ : $P\to {ℝ}^{n}$, $\eta \ne 0$, such that $f\left(x\right)-f\left({x}_{0}\right)\ge {\left[{\nabla }_{x}f\left({x}_{0}\right)\right]}^{T}\eta \left(x\right)$ and $-g\left({x}_{0}\right)\ge {\left[{\nabla }_{x}g\left({x}_{0}\right)\right]}^{T}\eta \left(x\right)$ 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 Free problems in several independent variables (optimality conditions) 49N15 Duality theory (optimization) 90C30 Nonlinear programming
##### References:
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