Constrained Saddle Point Problems Research Articles

Uzawa-type Iterative Solution Methods for Constrained Saddle Point Problems

For finite-dimensional saddle point problem with a nonlinear monotone operator and constraints on direct variables, iterative methods are developed, which in the potential case can be viewed as preconditioned Uzawa methods or as Uzawa-block relaxation methods. Convergence conditions of the iterative methods are formulated in the form of operator inequalities connecting the operator of the problem and the preconditioning matrix. When applied to mesh problems, this allows us to construct suitable preconditioners that ensure the convergence and effective implementation of iterative methods and to obtain the admissible intervals of iterative parameters which don’t depend on mesh parameters. The presented results are based on the general theory developed by the author with co-authors in recent years.

Domain decomposition and Uzawa-type iterative method for elliptic variational inequality

Finite element approximation of an elliptic variational inequality with quaisilinear operator and constraints to the solution and its gradient is constructed. Corresponding discrete problem is splitted into subproblems by non-overlapping domain decomposition technique and constrained saddle point problem is constructed for the splitted problem. Block relaxation-Uzawa iterative solution method is applied to this resulting saddle point problem. Existence of a solution to saddle point problem and convergence of the iterative method are proved.

Non-overlapping domain decomposition method for a variational inequality with gradient constraints

Non-overlapping domain decomposition method is applied to a variational inequality with nonlinear diffusion-convection operator and gradient constraints. The method is based on the initial approximation of the problem and its subsequent splitting into subproblems. For the resulting constrained saddle point problem block relaxation-Uzawa iterative solution method is applied.

Efficiently implementable iterative methods for linear elliptic variational inequalities with constraints on the gradient of solution

We construct and investigate a new iterative solution method for a finite-dimensional constrained saddle point problem. The results are applied to prove the convergence of different iterativemethods formesh approximations of variational inequalities with constraints on the gradient of solution. In particular, we prove the convergence of two-stage iterative methods. The main advantage of the proposed methods is the simplicity of their implementation. The numerical testing demonstrates high convergence rate of the methods.

Easily implementable iterative methods for variational inequalities with nonlinear diffusion–convection operator and constraints to the gradient of solution

AbstractNew iterative solution methods are proposed for the finite element approximation of a class of variational inequalities with nonlinear diffusion-convection operator and constraints to the gradient of solution. Implementation of every iteration of these methods reduces to the solution of a system of linear equations and a set of two-dimensional minimization problems. Convergence is proved by the application of a general result on the convergence of the iterative methods for a nonlinear constrained saddle point problem.

Iterative solution methods for mesh approximation of control and state constrained optimal control problem with observation in a part of the domain

Iterative solution methods for finite-dimensional constrained saddle point problems are investigated theoretically and numerically. These saddle point problems arise when approximating differential optimal control problems with point-wise state and control constraints by finite element or finite difference schemes with further using Lagrange multipliers technique. The linear elliptic boundary value problems with distributed control and the observation in a part of the domain are considered. Equivalent transformations of the constructed finite-dimensional saddle point problem are executed to apply effectively Uzawa-type iterative methods. Numerical comparison of these methods with gradient method for a regularized problem and interior point method is done.

An efficient iterative scheme for the highly constrained augmented Stokes problem for the numerical simulation of flows in porous media

In this work, we present a new efficient iterative solution technique for large sparse matrix systems that are necessary in the mixed finite-element formulation for flow simulations of porous media with complex 3D architectures in a representative volume element. Augmented Stokes flow problems with the periodic boundary condition and the immersed solid body as constraints have been investigated, which form a class of highly constrained saddle point problems mathematically. By solving the generalized eigenvalue problem based on block reduction of the discrete systems, we investigate structures of the solution space and its subspaces and propose the exact form of the block preconditioner. The exact Schur complement using the fundamental solution has been proposed to implement the block-preconditioning problem with constraints. Additionally, the algebraic multigrid method and the diagonally scaled conjugate gradient method are applied to the preconditioning sub-block system and a Krylov subspace method (MINRES) is employed as an outer solver. We report the performance of the present solver through example problems in 2D and 3D, in comparison with the approximate Schur complement method. We show that the number of iterations to reach the convergence is independent of the problem size, which implies that the performance of the present iterative solver is close to O(N).

Iterative solution methods for variational inequalities with nonlinear main operator and constraints to gradient of solution

Convergence of the preconditioned Uzawa-type and Arrow-Hurwitz-type iterative methods for nonlinear finite dimensional constrained saddle point problems is investigated. The general results are applied to the finite element approximation of a variational inequality with nonlinear main operator and constraints to gradient of solution.

Preconditioned Uzawa-type methods for finite-dimensional constrained saddle point problems

A new approach for the constructing and investigating the algorithms to solve a class of finite dimensional constrained saddle point problems is suggested. General convergence theorem for the preconditioned Uzawa-type iterative method is proved and applied to the finite element approximations of variational inequalities and optimal control problems.

An interior point method for constrained saddle point problems

We present an algorithm for the constrained saddle point problem with a convex-concave function L and convex sets with nonempty interior. The method consists of moving away from the current iterate by choosing certain perturbed vectors. The values of gradients of L at these vectors provide an appropriate direction. Bregman functions allow us to define a curve which starts at the current iterate with this direction, and is fully contained in the interior of the feasible set. The next iterate is obtained by moving along such a curve with a certain step size. We establish convergence to a solution with minimal conditions upon the function L, weaker than Lipschitz continuity of the gradient of L, for instance, and including cases where the solution needs not be unique. We also consider the case in which the perturbed vectors are on certain specific curves starting at the current iterate, in which case another convergence proof is provided. In the case of linear programming, we obtain a family of interior point methods where all the iterates and perturbed vectors are computed with very simple formulae, without factorization of matrices or solution of linear systems, which makes the method attractive for very large and sparse matrices. The method may be of interest for massively parallel computing. Numerical examples for the linear programming case are given.

An algorithm for a constrained saddle point problem via an exact penalty method

In this paper we present an algorithm which allows the approximation of a constrained saddle point problem by a sequence of unconstrained and finite dimensional optimization problems using an exact penalty functional for constrained minimax problems. Numerical experiments, both in finite and infinite dimension, are reported and are compared with those obtained by the use of an external penalization.

Convergence results of an approximation method for constrained saddle point problems

This paper concerns algorithms allowing numerical approximations of the solution of constrained saddle point problems by a sequence of finite dimensional unconstrained optimization problems. The obtained results are presented in two papers. The first one is on the theoretical results: strong convergence in Hilbert space for strongly convex- concave objective functional is proved, and an improvement in the finite dimensional case is obtained. In a following paper [3] numerical results are presented in both finite and infinite dimensional case (with application to optimal control theory) and they show the validity of the presented algorithms.

Implementation and numerical results of an approximation method for constrained saddle point problems

This paper completes the study made in a preceding article [2] on algorithms allowing numerical approximations of the solution of constrained saddle point problems by a sequence of finite dimensional unconstrained optimization problems. In that paper strong convergence in Hilbert space for strongly convex-concave objective functionals is proved and an improvement in the finite dimensional case is obtained. In this paper, application to the optimal control theory is made and numerical results are presented in both the finite and infinite dimensional case which show the validity of the presented algorithms.

Equivalence in nonlinear programming

AbstractThe purpose of this work is to show the relation between the Kuhn‐Tucker and the gradient‐projection conditions for constrained maximization (minimization) problems. A similar relationship exists between Kuhn‐Tucker‐type conditions and gradient‐projection‐type conditions for certain constrained saddle‐point problems. This latter relationship may lead to a computational algorithm for these constrained saddle‐point problems.