We propose a new distributed algorithm for empirical risk minimization in machine learning. The algorithm is based on an inexact damped Newton method, where the inexact Newton steps are computed by a distributed preconditioned conjugate gradient method. We analyze its iteration complexity and communication efficiency for minimizing self-concordant empirical loss functions, and discuss the resul...

We show how certain widely used multistep approximation algorithms can be interpreted as instances of an approximate Newton method. It was shown in an earlier paper by the second author that the convergence rates of approximate Newton methods (in the context of the numerical solution of PDEs) suuer from a \loss of derivatives", and that the subsequent linear rate of convergence can be improved ...

In this letter, we present a computational framework based on the use of the Newton and level set methods and tailored for the modeling of bubbles with surface tension in a surrounding Newtonian fluid. We describe a fully implicit and monolithic finite element method that maintains stability for significantly larger time steps compared to the usual explicit method and features substantial compu...

We present a method for solving polynomial equations over idempotent ω-continuous semirings. The idea is to iterate over the semiring of functions rather than the semiring of interest, and only evaluate when needed. The key operation is substitution. In the initial step, we compute a linear completion of the system of equations that exhaustively inserts the equations into one another. With func...

Newton’s method is one of the most famous numerical methods. Its origins, as the name suggests, lies in part with Newton, but the form familiar to us today is due to Simpson of “Simpson’s Rule” fame. Originally proposed to find the roots of polynomials it was Simpson who proposed using it for general nonlinear equations and for its use in optimization by finding a root of the gradient. Here we ...

We study an infeasible interior path-following method for complemen-tarity problems. The method uses a wide neighborhood and takes one or two Newton steps per iteration. We show that the method attains global convergence, assuming the iterates are deened and bounded (which occurs when the function is a P 0-R 0-function or when the function is monotone and a suuciently positive solution exists)....

By making use of duality mappings, we formulate an inexact Newton– Landweber iteration method for solving nonlinear inverse problems in Banach spaces. The method consists of two components: an outer Newton iteration and an inner scheme providing the increments by applying the Landweber iteration in Banach spaces to the local linearized equations. It has the advantage of reducing computational w...

Solutions of several problems can be modelled as solutions of nonsmooth equations. Then, Newton-type methods for solving such equations induce particular iteration steps (actions) and regularity requirements in the original problems. We study these actions and requirements for nonlinear complementarity problems (NCP’s) and Karush–Kuhn–Tucker systems (KKT) of optimization models. We demonstrate ...

This paper is concerned with the development of a fully implicit and purely Eulerian fluid-structure interaction method tailored for the modeling of the large deformations of elastic membranes in a surrounding Newtonian fluid. We consider a simplified model for the mechanical properties of the membrane, in which the surface strain energy depends on the membrane stretching. The fully Eulerian de...

We propose a semismooth active-set Newton algorithm for solving the nonlinear complementarity problems with degenerate solutions. This method introduces the active-set technique to identify the degenerate set. At each iteration, the search direction is obtained by two reduced linear systems. Instead of employing gradient steps as adjustments to guarantee the sufficient reduction of the merit fu...