In part I of this article, we proposed a Lagrange–Newton–Krylov–Schur (LNKS) method for the solution of optimization problems that are constrained by partial differential equations. LNKS uses Krylov iterations to solve the linearized Karush–Kuhn–Tucker system of optimality conditions in the full space of states, adjoints, and decision variables, but invokes a preconditioner inspired by reduced ...

We analyse the elementwise convergence in the Jacobian of the classical Newton method as the solution of a non-linear system f(x) = 0 is approached. It is shown that each Jacobian element approaches a constant value at a rate determined by the nonlinearity of the relevant function with respect to the appropriate scalar variable. For many practical problems, this means that in the last steps of ...

The randomized subspace Newton convex methods for the sensor selection problem are proposed. algorithm is straightforwardly applied to formulation, and customized method in which part of update variables selected be present best candidates also considered. In converged solution, almost same results obtained by original randomized-subspace-Newton methods. As expected, require more computational ...

In this paper we study new preconditioners to be used within the Nonlinear Conjugate Gradient (NCG) method, for large scale unconstrained optimization. The rationale behind our proposal draws inspiration from quasi– Newton updates, and its aim is to possibly approximate in some sense the inverse of the Hessian matrix. In particular, at the current iteration of the NCG we consider some precondit...

Optimization using the L∞ norm is an increasingly important area in multiview geometry. Previous work has shown that globally optimal solutions can be computed reliably using the formulation of generalized fractional programming, in which algorithms solve a sequence of convex problems independently to approximate the optimal L∞ norm error. We found the sequence of convex problems are highly rel...

We give conditions under which limited-memory quasi-Newton methods with exact line searches will terminate in n steps when minimizing n-dimensional quadratic functions. We show that although all Broyden family methods terminate in n steps in their full-memory versions, only BFGS does so with limited-memory. Additionally, we show that full-memory Broyden family methods with exact line searches t...

We give a framework for the globalization of a nonsmooth Newton method. In part one we start with recalling B. Kummer's approach to convergence analysis of a nonsmooth Newton method and state his results for local convergence. In part two we give a globalized version of this method. Our approach uses a path search idea to control the descent. After elaborating the single steps, we analyze and p...

A general outer iteration, based upon linearization, is introduced at discrete time steps for the one-dimensional semiconductor device model. The iteration depends upon solving the semidiscrete device equations approximately, specifically, in such a way that the residual is of order ∆t in an appropriate norm. It is shown that this maintains the order of the backward Euler method. A monitoring o...

Most existing methods for calculating the steady state solution of the lattice Boltzmann equations are based on pseudo time stepping, which often requires a large number of time steps especially for high Reynolds number problems. To calculate the steady state solution directly without the time integration, in this paper we propose and study a nonlinearly preconditioned inexact Newton algorithm ...

Current successful methods for solving semidefinite programs, SDP, are based on primal-dual interior-point approaches. These usually involve a symmetrization step to allow for application of Newton’s method followed by block elimination to reduce the size of the Newton equation. Both these steps create ill-conditioning in the Newton equation and singularity of the Jacobian of the optimality con...