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Keywords:
vector optimization; locally Lipschitz optimization; Dini derivatives; optimality conditions
vector optimization; locally Lipschitz optimization; Dini derivatives; optimality conditions
Summary:
The present paper studies the following constrained vector optimization problem: $\min _Cf(x)$, $g(x)\in -K$, $h(x)=0$, where $f\colon\Bbb R^n\to \Bbb R^m$, $g\colon\Bbb R^n\to \Bbb R^p$ are locally Lipschitz functions, $h\colon\Bbb R^n\to \Bbb R^q$ is $C^1$ function, and $C\subset \Bbb R^m$ and $K\subset \Bbb R^p$ are closed convex cones. Two types of solutions are important for the consideration, namely $w$-minimizers (weakly efficient points) and $i$-minimizers (isolated minimizers of order 1). In terms of the Dini directional derivative first-order necessary conditions for a point $x^0$ to be a $w$-minimizer and first-order sufficient conditions for $x^0$ to be an $i$-minimizer are obtained. Their effectiveness is illustrated on an example. A comparison with some known results is done.
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