Some Quadrature-Based Versions of the Generalized Newton Method for Solving Unconstrained Optimization Problems

Abstract

Many practical problems require solving system of nonlinear equations. A lot of optimization problems can be transformed to the equation $F(x)=0$, where the function F is nonsmooth, i.e. nonlinear complementarity problem or variational inequality problem. We propose a modifications of a generalized Newton method based on some rules of quadrature. We consider these algorithms for solving unconstrained optimization problems, in which the objective function is only LC$^{1}$, i.e. has not differentiable gradient. Such problems often appear in nonlinear programming, usually as subproblems. The methods considered are Newton-like iterative schemes, however they use combination of elements of some subdifferential. The methods are locally and at least superlinearly convergent under mild conditions imposed on the gradient of the objective function. Finally, results of numerical tests are presented.