Locally Lipschitz vector optimization with inequality and equality constraints

Abstract

The present paper studies the following constrained vector optimization problem: \(\mathop {\min }\limits_C f(x),g(x) \in - K,h(x) = 0\), where f: ℝn → ℝm, g: ℝn → ℝp are locally Lipschitz functions, h: ℝn → ℝq is C 1 function, and C ⊂ ℝm and K ⊂ ℝp 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.