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Computation of generalized differentials in nonlinear complementarity problems

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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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Correspondence to Shuhuang Xiang.

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Dedicate to Professor Liqun Qi on his 65 birthday.

This paper is supported partly by the Hong Kong Research Grant Council, NSF of China (Nos. 10771218, 11071260) and the Program for New Century Excellent Talents in University, State Education Ministry, China.

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Xiang, S., Chen, X. Computation of generalized differentials in nonlinear complementarity problems. Comput Optim Appl 50, 403–423 (2011). https://doi.org/10.1007/s10589-010-9349-z

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