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Newton’s method for the nonlinear complementarity problem: a B- differentiable equation approach. (English) Zbl 0724.90071
The nonlinear complementarity problem $$F^ T(x)x=0$$, $$F(x)\in R^ n_+$$ can be equivalently formulated as a system of nonlinear equations $$H(x)=\min (x,F(x))=0$$, where the ‘min’ operation is taken componentwise. The function H is not Fréchet differentiable in general, however, it is B-differentiable. S. Robinson was the first to study Newton’s method for equations with such functions. The present paper develops a previous work by the first author and J.-S. Pang [in: Computational solution of nonlinear systems of equations, Proc. SIAM-AMS Summer Semin., Ft. Collins/CO (USA) 1988, Lect. Appl. Math. 26, 265-284 (1990; Zbl 0699.65054)] converting the original problem into a system of equations through the use of a Minty map. At each step of the algorithm a linear system is solved and a line search for a given merit function is performed. Numerical results and a comparison with the traditional Josephy-Newton method are presented.

##### MSC:
 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 49J50 Fréchet and Gateaux differentiability in optimization 65K05 Numerical mathematical programming methods 90C30 Nonlinear programming
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