Computation of generalized differentials in nonlinear complementarity problems

Shuhuang Xiang, Xiaojun Chen

Research output: Journal article publicationJournal articleAcademic researchpeer-review

2 Citations (Scopus)


Let f and g be continuously differentiable functions on ℝn. 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).
Original languageEnglish
Pages (from-to)403-423
Number of pages21
JournalComputational Optimization and Applications
Issue number2
Publication statusPublished - 1 Oct 2011


  • B-differential
  • Clarke generalized Jacobian
  • Complementarity problem
  • Error bound
  • Fréchet differential
  • Generalized Newton method
  • NCP-function

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Control and Optimization


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