In the paper, a Newton-type method for solution of generalized equations (GEs) is derived, where linearization concerns both single-valued and multivalued part considered GE. The based on new notion semismoothness${}^*$, which, together with suitable regularity condition, ensures local superlinear convergence. An implementable version derived class GEs, frequently arising in optimization equili...

We develop general approximate Newton methods for solving Lips-chitz continuous equations by replacing the iteration matrix with a consistently approximated Jacobian thereby reducing the computation in the generalized Newton method. Locally superlinear convergence results are presented under moderate assumptions. To construct a consistently approximated Jacobian, we introduce two main methods: ...

Variational inequalities, nonlinear programming, complementarity problems and other problems can be reduced to nonsmooth equations, for which some generalizations of Newton's method are known. The Newton path, as a natural generalization of the Newton direction, was suggested by D. Ralph for enlarging the convergence region (globalization) of Newton-Robinson's method in the nonsmooth case. We i...

Let P ,X and Y be Banach spaces. Suppose that f : P ×X → Y is continuously Fréchet differentiable function depend on the point (p, x) and F : X ⇒ 2 is a set-valued mapping with closed graph. Consider the following parametric generalized equation of the form: 0 ∈ f(p, x) + F (x). (1) In the present paper, we study an extended Newton-type method for solving parametric generalized equation (1). In...

Newton-Krylov methods, primarily using the Jacobian-Free Newton-Krylov (JFNK) approximation, are examined as an alternative to the traditional power iteration method for the calculation of the fundamental eigenmode in reactor analysis applications based on diffusion theory. One JFNK approach can be considered an acceleration technique for the standard power iteration as it is “wrapped around” t...

in this paper, we study the problem of minimizing the ratio of two quadratic functions subject to a quadratic constraint. first we introduce a parametric equivalent of the problem. then a bisection and a generalized newton-based method algorithms are presented to solve it. in order to solve the quadratically constrained quadratic minimization problem within both algorithms, a semidefinite optim...