113 research outputs found
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
An asymptotically superlinearly convergent semismooth Newton augmented Lagrangian method for Linear Programming
Powerful interior-point methods (IPM) based commercial solvers, such as
Gurobi and Mosek, have been hugely successful in solving large-scale linear
programming (LP) problems. The high efficiency of these solvers depends
critically on the sparsity of the problem data and advanced matrix
factorization techniques. For a large scale LP problem with data matrix
that is dense (possibly structured) or whose corresponding normal matrix
has a dense Cholesky factor (even with re-ordering), these solvers may require
excessive computational cost and/or extremely heavy memory usage in each
interior-point iteration. Unfortunately, the natural remedy, i.e., the use of
iterative methods based IPM solvers, although can avoid the explicit
computation of the coefficient matrix and its factorization, is not practically
viable due to the inherent extreme ill-conditioning of the large scale normal
equation arising in each interior-point iteration. To provide a better
alternative choice for solving large scale LPs with dense data or requiring
expensive factorization of its normal equation, we propose a semismooth Newton
based inexact proximal augmented Lagrangian ({\sc Snipal}) method. Different
from classical IPMs, in each iteration of {\sc Snipal}, iterative methods can
efficiently be used to solve simpler yet better conditioned semismooth Newton
linear systems. Moreover, {\sc Snipal} not only enjoys a fast asymptotic
superlinear convergence but is also proven to enjoy a finite termination
property. Numerical comparisons with Gurobi have demonstrated encouraging
potential of {\sc Snipal} for handling large-scale LP problems where the
constraint matrix has a dense representation or has a dense
factorization even with an appropriate re-ordering.Comment: Due to the limitation "The abstract field cannot be longer than 1,920
characters", the abstract appearing here is slightly shorter than that in the
PDF fil
Merit functions: a bridge between optimization and equilibria
In the last decades, many problems involving equilibria, arising from engineering, physics and economics, have been formulated as variational mathematical models. In turn, these models can be reformulated as optimization problems through merit functions. This paper aims at reviewing the literature about merit functions for variational inequalities, quasi-variational inequalities and abstract equilibrium problems. Smoothness and convexity properties of merit functions and solution methods based on them will be presented
On a Nonsmooth Gauss–Newton Algorithms for Solving Nonlinear Complementarity Problems
In this paper, we propose a new version of the generalized damped Gauss–Newton method for solving nonlinear complementarity problems based on the transformation to the nonsmooth equation, which is equivalent to some unconstrained optimization problem. The B-differential plays the role of the derivative. We present two types of algorithms (usual and inexact), which have superlinear and global convergence for semismooth cases. These results can be applied to efficiently find all solutions of the nonlinear complementarity problems under some mild assumptions. The results of the numerical tests are attached as a complement of the theoretical considerations
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