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On duality for nonsmooth lipschitz optimization problems

We present some duality theorems for a non-smooth Lipschitz vector optimization problem. Under generalized invexity assumptions on the functions the duality theorems do not require constraint qualifications. | Yugoslav Journal of Operations Research Vol 19 (2009), Number 1, 41-47 DOI: 10.2298/YUJOR0901041P ON DUALITY FOR NONSMOOTH LIPSCHITZ OPTIMIZATION PROBLEMS Vasile PREDA University of Bucharest, Bucharest preda@fmi.unibuc.ro Miruna BELDIMAN Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Bucharest Anton BĂTĂTORESCU University of Bucharest, Bucharest Received: December 2007 / Accepted: June 2009 Abstract: We present some duality theorems for a non-smooth Lipschitz vector optimization problem. Under generalized invexity assumptions on the functions the duality theorems do not require constraint qualifications. Keywords: Nonsmooth Lipschitz vector optimization, Fritz John type necessary optimization conditions, duality theorems. 1. INTRODUCTION We shall introduce some definitions used in this article and formulate a vector optimization problem together with its Mond-Weir dual. The real n-dimensional vector space will be denoted by R n and we will use the following conventions for any two vectors x, y ∈ R n : x 0 such that for any y, z ∈ N ( x ) we have ϕ ( y) − ϕ ( z) ≤ K x y − z . Definition 1.2 (Clarke [1]) The generalized directional derivative of a local Lipschitz function ϕ at x in the direction d is denoted by ϕ o ( x; d ) = lim sup y→ x t 20 ϕ ( y + td ) − ϕ ( y ) t . Definition 1.3 The Clarke generalized subgradient of a locally Lipschitz function ϕ at x is denoted by ∂ cϕ ( x) = {ξ ∈ X ∗ | ϕ o ( x; d ) ≥ ξ , d , ∀d ∈ X } . Definition 1.4 (see also Giorgi and Guerraggio [2]) Let us consider: η : X × X → X , ρ ∈ R, d : X × X → R + . We say that: ϕ is (η , ρ ) -pseudoinvex if for ∀x, y ∈ X , ϕ o ( x;η ( y, x )) ≥ ρ d ( y, x) ⇒ ϕ ( y ) ≥ ϕ ( x), or, equivalently, for ∀x, y ∈ X , ∀ξ ∈ ∂ cϕ ( x ), ϕ ( y ) ϕ ( x), V. Preda, M. Beldiman, A.Bătătorescu / On Duality for Nonsmooth Lipschitz 43 or, equivalently, for ∀x, y ∈ X , with x ≠ y , and ∀ξ ∈ ∂ cϕ ( x ) , ϕ ( y ) ≤ ϕ ( x) ⇒ ξ ,η ( y, x) 0} . From (0.2) we have g j .