Nonsmooth generalized complementarity as unconstrained optimization

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April 2010, 6(2): 411-423. doi: 10.3934/jimo.2010.6.411

Nonsmooth generalized complementarity as unconstrained optimization

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

Received March 2009 Revised January 2010 Published March 2010

Abstract
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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.

Keywords: Merit function, GCP function, semismooth-functions, locally Lipschitzian, generalized Jacobian, $H$-differentiability., generalized complementarity problem.

Mathematics Subject Classification: Primary: 65K05, 90C33; Secondary: 49J5.

Citation: Mohamed Aly Tawhid. Nonsmooth generalized complementarity as unconstrained optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 411-423. doi: 10.3934/jimo.2010.6.411

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