Computation of generalized differentials in nonlinear complementarity problems

  • Authors:

  • Shuhuang Xiang;Xiaojun Chen

  • Affiliations:

  • Department of Applied Mathematics and Software, Central South University, Changsha, PR China 410083;Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

  • Venue:
  • Computational Optimization and Applications
  • Year:
  • 2011

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Abstract

Let f and g be continuously differentiable functions on R n . The nonlinear complementarity problem NCP(f,g), 0驴f(x)驴g(x)驴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).