Deflation for semismooth equations

Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea is the combination of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, deflation constructs a new problem for which a semismooth Newton method will not converge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth Newton method. We demonstrate its utility on various finite- and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics.Comment: 24 pages, 3 figure

Hybrid Newton-type method for a class of semismooth equations

In this paper, we present a hybrid method for the solution of a class of composite semismooth equations encountered frequently in applications. The method is obtained by combining a generalized finite-difference Newton method to an inexpensive direct search method. We prove that, under standard assumptions, the method is globally convergent with a local rate of convergence which is superlinear or quadratic. We report also several numerical results obtained applying the method to suitable reformulations of well-known nonlinear complementarity problem

Newton methods for solving two classes of nonsmooth equations

summary:The paper is devoted to two systems of nonsmooth equations. One is the system of equations of max-type functions and the other is the system of equations of smooth compositions of max-type functions. The Newton and approximate Newton methods for these two systems are proposed. The Q-superlinear convergence of the Newton methods and the Q-linear convergence of the approximate Newton methods are established. The present methods can be more easily implemented than the previous ones, since they do not require an element of Clarke generalized Jacobian, of B-differential, or of b-differential, at each iteration point

A UNIFIED INTERIOR POINT FRAMEWORK FOR OPTIMIZATION ALGORITHMS

Interior Point algorithms are optimization methods developed over the last three decades following the 1984 fundamental paper of Karmarkar. Over this period, IPM algorithms have had a profound impact on optimization theory as well as practice and have been successfully applied to many problems of business, engineering and science. Because of their operational simplicity and wide applicability, IPM algorithms are now playing an increasingly important role in computational optimization and operations research. This article provides unified interior point algorithms to optimization problems as well as comparing performances with classical algorithms. Keywords Interior Point methods, Optimization algorithms, Lagrangian Multipliers,  Barrier methods, Newton’s method, Matrix-free method

Truncated Nonsmooth Newton Multigrid for phase-field brittle-fracture problems

We propose the Truncated Nonsmooth Newton Multigrid Method (TNNMG) as a solver for the spatial problems of the small-strain brittle-fracture phase-field equations. TNNMG is a nonsmooth multigrid method that can solve biconvex, block-separably nonsmooth minimization problems in roughly the time of solving one linear system of equations. It exploits the variational structure inherent in the problem, and handles the pointwise irreversibility constraint on the damage variable directly, without penalization or the introduction of a local history field. Memory consumption is significantly lower compared to approaches based on direct solvers. In the paper we introduce the method and show how it can be applied to several established models of phase-field brittle fracture. We then prove convergence of the solver to a solution of the nonsmooth Euler-Lagrange equations of the spatial problem for any load and initial iterate. Numerical comparisons to an operator-splitting algorithm show a speed increase of more than one order of magnitude, without loss of robustness