Nonsmooth analysis of vector-valued mappings with contributions to nondifferentiable programming

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Abstract
We define order Lipschitz mappings from a Banach space to an order complete vector lattice and present a nonsmooth analysis for such functions. In particular, we establish properties of a generalized directional derivative and gradient and derive results concerning a calculus of generalized gradients (i.e., calculation of the generalized gradient of f when f = f1 + f2, f = f · 2, etc.). We show the relevance of the above analysis to nondifferentiaile programming by deriving optimality conditions for problems of the form min f(x) subject to x [euro] S. For S arbitrary we state the results in terms of cones of displacement of the feasible region at the optimal point; when S ={x ∊ A|g(x) ∊ B}, we obtain Kuhn-Tucker type results.

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