Computation of generalized differentials in nonlinear complementarity problems |
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Authors: | Shuhuang Xiang Xiaojun Chen |
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Institution: | 1.Department of Applied Mathematics and Software,Central South University,Changsha,PR China;2.Department of Applied Mathematics,The Hong Kong Polytechnic University,Kowloon,Hong Kong |
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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). |
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