Abstract
Abstract In this paper, we define new vector generalized convexity, namely nondifferentiable vector ( G f , β f ) -invexity, for a given locally Lipschitz vector function f. Basing on this new nondifferentiable vector generalized invexity, we have managed to deal with nondifferentiable nonlinear programming problems under some assumptions. Firstly, we present G-Karush-Kuhn-Tucker necessary optimality conditions for nonsmooth mathematical programming problems. With the new vector generalized invexity assumption, we also obtain G-Karush-Kuhn-Tucker sufficient optimality conditions for the same programming problems. Moreover, we establish duality results for this kind of multiobjective programming problems. In the end, a suitable example illustrates that the new optimality results are more useful for some class of optimization problems than the optimality conditions with invex functions. MSC:90C26.
Highlights
Convexity plays a central role in many aspects of mathematical programming including the analysis of stability, sufficient optimality conditions and duality
Invexity, was introduced by Hanson in [ ]. He has shown that invexity has a common property in mathematical programming with convexity and that Karush-KuhnTucker conditions are sufficient for global optimality of nonlinear programming under the invexity assumptions
With (G-considered nonsmooth multiobjective programming problem (CVP)), we have proved the G-Karush-Kuhn-Tucker necessary optimality conditions for (CVP)
Summary
Convexity plays a central role in many aspects of mathematical programming including the analysis of stability, sufficient optimality conditions and duality. Definition A feasible point xis said to be a (weakly) efficient solution for a multiobjective programming problem (CVP) if and only if there exists no x ∈ E such that f (x) ≤ (
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