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Locally Lipschitz vector optimization with inequality and equality constraints. (English) Zbl 1224.90154
Summary: The present paper studies the following constrained vector optimization problem: \(\min _Cf(x)\), \(g(x)\in -K\), \(h(x)=0\), where \(f\:\mathbb R^n\to \mathbb R^m\), \(g\:\mathbb R^n\to \mathbb R^p\) are locally Lipschitz functions, \(h\:\mathbb R^n\to \mathbb R^q\) is \(C^1\) function, and \(C\subset \mathbb R^m\) and \(K\subset \mathbb R^p\) 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.

MSC:
90C29 Multi-objective and goal programming
90C30 Nonlinear programming
90C46 Optimality conditions and duality in mathematical programming
49J52 Nonsmooth analysis
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