Global and local superlinear convergence analysis of Newton-type methods for semismooth equations with smooth least squares.

*(English)*Zbl 0928.65060
Fukushima, Masao (ed.) et al., Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods. Session in the 16th international symposium on Mathematical programming (ismp97) held at Lausanne EPFL, Switzerland, August 24–29, 1997. Boston: Kluwer Academic Publishers. Appl. Optim. 22, 181-209 (1999).

Summary: The local superlinear convergence of the generalized Newton method for solving systems of nonsmooth equations has been proved by L. Qi and J. Sun [Math. Program. 58A, No. 3, 353-367 (1993; Zbl 0780.90090)] under the semismooth condition and nonsingularity of the generalized Jacobian at the solution. Unlike the Newton method for systems of smooth equations, globalization of the generalized Newton method seems difficult to achieve in general. However, we show that global convergence analysis of various traditional Newton-type methods for systems of smooth equations can be extended to systems of nonsmooth equations with semismooth operators whose least squares objective is smooth.

The value of these methods is demonstrated from their applications to various semismooth equation reformulation of nonlinear complementarity and related problems.

For the entire collection see [Zbl 0909.00046].

The value of these methods is demonstrated from their applications to various semismooth equation reformulation of nonlinear complementarity and related problems.

For the entire collection see [Zbl 0909.00046].

##### MSC:

65H10 | Numerical computation of solutions to systems of equations |

65K05 | Numerical mathematical programming methods |

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

90C53 | Methods of quasi-Newton type |