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Author Country (or Countries)

Egypt

Abstract

For H-differentiable function f from a closed rectangle Q in Rn into Rn, a result of Song, Gowda and Ravindran [On Characterizations of P- and P0-Properties in Nonsmooth Functions. Mathematics of Operations Research. 25: 400-408 (2000)] asserts that f is a P(P0)− function on Q if the HQ-differential TQ(x) at each x ∈ Q consisting of P(P0)− matrices. In this paper, we introduce the concepts of relatively P(P0)− properties in order to extend these results to nonsmooth functions when the underlying functions are H-differentiable.We give characterizations of relatively P(P0)− of vector nonsmooth functions. Also, our results give characterizations of relatively P(P0)− when the underlying functions are C1-functions, semismooth-functions, and for locally Lipschitzian functions. Moreover, we show useful applications of our results by giving illustrations to generalized complementarity problems.

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