We develop a globally convergent algorithm based on the LP-Newton method which has been recently proposed for solving constrained equations, possibly nonsmooth and possibly with nonisolated solutions. The new algorithm makes use of linesearch for the natural merit function, and preserves the strong local convergence properties of the original LP-Newton scheme. We also present computational expe...

We develop a globally convergent algorithm based on the LP-Newton method, which has been recently proposed for solving constrained equations, possibly nonsmooth and possibly with nonisolated solutions. The new algorithm makes use of linesearch for the natural merit function and preserves the strong local convergence properties of the original LP-Newton scheme. We also present computational expe...

While generalized equations with differentiable single-valued base mappings and the associated Josephy–Newton method have been studied extensively, the setting with semismooth base mapping had not been previously considered (apart from the two special cases of usual nonlinear equations and of Karush-Kuhn-Tucker optimality systems). We introduce for the general semismooth case appropriate notion...

A class of semismooth Newton methods for unilaterally constrained variational problems modelling cracks under a non-penetration condition are introduced and investigated. On the continuous level, a penalization technique is applied which allows to argue generalized differentiability of the nonlinear mapping associated to its first order optimality characterization. It is shown that the correspo...

Solving systems of nonlinear equations is perhaps one of the most difficult problems in all numerical computation. Although numerous methods have been developed to attack this class of numerical problems, one of the simplest and oldest methods, Newton’s method is arguably the most commonly used. As is well known, the convergence and performance characteristics of Newton’s method can be highly s...

In this paper, we focus on fractional programming problems that minimize the ratio of two indefinite quadratic functions subject to two quadratic constraints. Utilizing the relationship between fractional programming and parametric programming, we transform the original problem into a univariate nonlinear equation. To evaluate the function in the equation, we need to solve a problem of minimizi...

More than a decade agao, Newton’s method has been proposed for constructing the convex best interpolant. Its local quadratic convergence has only been established recently by recasting it as the generalized Newton method for semismooth equations. It still remains mysterious that the Newton method coupled with line search strategies works practically well in global sense. Similar to the classica...

We devise a new generalized univariate Newton method for solving nonlinear equations, motivated by Bregman distances and proximal regularization of optimization problems. We prove quadratic convergence of the new method, a special instance of which is the classical Newton’s method. We illustrate the possible benefits of the new method over classical Newton’s method by means of test problems inv...

Based on the identification of indices active at a solution of the mixed complementarity problem (MCP), we propose a class of Newton methods for which local superlinear convergence holds under extremely mild assumptions. In particular, the error bound condition needed for the identification procedure and the nondegeneracy condition needed for the convergence of the resulting Newton method are i...

We present a generalized Newton method and a quasiNewton method for solving H(x) := F(nc(x))+x-nc(x) = 0, when C is a polyhedral set. For both the Newton and quasi-Newton methods considered here, the subproblem to be solved is a linear system of equations per iteration. The other characteristics of the quasi-Newton method include: (i) a g-superlinear convergence theorem is established without a...