Newton’s method for B-differentiable equations.

*(English)*Zbl 0716.90090The author extends the classical Newton method for solving continuously differentiable systems of nonlinear equations to B-differentiable systems. The notion of B-differentiability was introduced by S. M. Robinson [Math. Program. Study 30, 45-66 (1987; Zbl 0629.90079)]. In general, given a B-differentiable function it is easy to state the analogue of the classical Newton method for finding a zero of the function. All what is required is to replace the F-derivative with the B- derivative. Several issues arise. First, what is the resulting Newton equation? How to solve it? Second, what are the convergence properties of the method? Third, is the method practically efficient? Does it give rise to some improved methods for nonlinear complementarity, variational inequality and nonlinear programming problems? The author provides answers to the questions raised above and a fundamental theory of Newton’s method.

Reviewer: G.Chen

##### MSC:

90C30 | Nonlinear programming |

49M15 | Newton-type methods |

49J40 | Variational inequalities |

65K05 | Numerical mathematical programming methods |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |