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Title: Computation of generalized differentials in nonlinear complementarity problems
Authors: Xiang, S
Chen, X 
Keywords: B-differential
Clarke generalized Jacobian
Complementarity problem
Error bound
Fréchet differential
Generalized Newton method
Issue Date: 2011
Publisher: Springer
Source: Computational optimization and applications, 2011, v. 50, no. 2, p. 403-423 How to cite?
Journal: Computational optimization and applications 
Abstract: Let f and g be continuously differentiable functions on ℝ n. The nonlinear complementarity problem NCP(f,g), 0≤f(x)⊥g(x)≥0, arises in many applications including discrete Hamilton-Jacobi-Bellman equations and nonsmooth Dirichlet problems. A popular method to find a solution of the NCP(f,g) is the generalized Newton method which solves an equivalent system of nonsmooth equations F(x)=0 derived by an NCP function. In this paper, we present a sufficient and necessary condition for F to be Fréchet differentiable, when F is defined by the "min" NCP function, the Fischer-Burmeister NCP function or the penalized Fischer-Burmeister NCP function. Moreover, we give an explicit formula of an element in the Clarke generalized Jacobian of F defined by the "min" NCP function, and the B-differential of F defined by other two NCP functions. The explicit formulas for generalized differentials of F lead to sharper global error bounds for the NCP(f,g).
ISSN: 0926-6003
EISSN: 1573-2894
DOI: 10.1007/s10589-010-9349-z
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