Journal ArticleDOI
On multilevel iterative methods for optimization problems
E. Gelman,Jan Mandel +1 more
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TLDR
The construction of auxiliary problems as well as applications to elasto-plastic model and linear programming are described and the auxiliary problem for the dual of a perturbed linear program is interpreted as a dual of perturbed aggregated linear program.Abstract:
This paper is concerned with multilevel iterative methods which combine a descent scheme with a hierarchy of auxiliary problems in lower dimensional subspaces. The construction of auxiliary problems as well as applications to elasto-plastic model and linear programming are described. The auxiliary problem for the dual of a perturbed linear program is interpreted as a dual of perturbed aggregated linear program. Coercivity of the objective function over the feasible set is sufficient for the boundedness of the iterates. Equivalents of this condition are presented in special cases.read more
Citations
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Journal ArticleDOI
Recursive Trust-Region Methods for Multiscale Nonlinear Optimization
TL;DR: A class of trust-region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations, leading to true multilevel/multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution ofpartial differential equations.
Journal ArticleDOI
Global and uniform convergence of subspace correction methods for some convex optimization problems
Xue-Cheng Tai,Jinchao Xu +1 more
TL;DR: Some global and uniform convergence estimates for a class of subspace correction (based on space decomposition) iterative methods applied to some unconstrained convex optimization problems are given.
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Adaptive multigrid methods for Signorini’s problem in linear elasticity
Ralf Kornhuber,Rolf Krause +1 more
TL;DR: In this paper, the authors derived globally convergent multigrid methods for the discretized Signorini problem in linear elasticity and observed similar convergence rates as for corresponding linear problems.
Journal ArticleDOI
Newton's method for the nonlinear complementarity problem: a B-differentiable equation problem
Patrick T. Harker,Baichun Xiao +1 more
TL;DR: A damped-Newton method for solving the nonlinear complementarity problem when it is formulated as a system of B-differentiable equations through the use of the Minty-map, which contains a one-dimensional line search and possesses a global convergence property under certain conditions.
References
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Book
Nonlinear Programming: Theory and Algorithms
TL;DR: The book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques.
Journal ArticleDOI
Multi-level adaptive solutions to boundary-value problems
TL;DR: In this paper, the boundary value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes, and interactions between these levels enable us to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); and conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining "°°-order" approximations and low n, even when singularities are present.
Book
A multigrid tutorial
TL;DR: This paper presents an implementation of Multilevel adaptive methods for Algebraic multigrid (AMG), a version of which has already been described in more detail in the preface.
Book
Methods and applications
TL;DR: A survey of sampling principles can be found in this paper, where the authors present sample designs for some common sampling problems such as bias and non-sampling errors in survey results.