Nonsmooth generalized complementarity asunconstrained optimization

Mohamed Aly Tawhid

doi:10.3934/jimo.2010.6.411

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2010, Volume 6, Issue 2: 411-423. Doi: 10.3934/jimo.2010.6.411

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Nonsmooth generalized complementarity as unconstrained optimization

Mohamed Aly Tawhid^1,

1.
Department of Mathematics and Statistics, Thompson Rivers University, 900 McGill Road, PO Box 3010, Kamloops, BC V2C 5N3

Received: February 28, 2009

Revised: December 31, 2009

Published: March 2010

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Abstract

We consider generalized complementarity problem GCP$(f,g)$ when the underlying functions $f$ and $g$ are $H$-differentiable. We describe $H$-differentials of some GCP functions and their merit functions. We give some conditions on the $H$-differentials of the given functions under which minimizing a merit function corresponding to such functions leads to a solution of the generalized complementarity problem. Further, we give some conditions on the functions $f$ and $g$ to get a solution of GCP$(f,g)$ by introducing the concepts of relative monotonicity and P₀-property and their variants. Our results further give a unified/generalization treatment of such results for the nonlinear complementarity problem when the underlying function is $C^1$ , semismooth, and locally Lipschitzian.

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Mathematics Subject Classification: Primary: 65K05, 90C33; Secondary: 49J50.

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