Zdeněk Dostál

Minimizing quadratic functions subject to bound constraints with the rate of convergence and finite termination

A new active set based algorithm is proposed that uses the conjugate gradient method to explore the face of the feasible region defined by the current iterate and the reduced gradient projection with the fixed steplength to expand the active set. The precision of approximate solutions of the auxiliary unconstrained problems is controlled by the norm of violation of the Karush-Kuhn-Tucker conditions at active constraints and the scalar product of the reduced gradient with the reduced gradient projection. The modifications were exploited to find the rate of convergence in terms of the spectral condition number of the Hessian matrix, to prove its finite termination property even for problems whose solution does not satisfy the strict complementarity condition, and to avoid any backtracking at the cost of evaluation of an upper bound for the spectral radius of the Hessian matrix. The performance of the algorithm is illustrated on solution of the inner obstacle problems. The result is an important ingredient in development of scalable algorithms for numerical solution of elliptic variational inequalities.

On a splitting type algorithm for the numerical realization of contact problems with Coulomb friction

This paper presents and analyses an iterative process for the numerical realization of contact problems with Coulomb friction which is based on the method of successive approximations combined with a splitting type approach. Numerical examples illustrate the efficiency of this method.

Duality-based domain decomposition with natural coarse-space for variational inequalities0

Abstract An efficient non-overlapping domain decomposition algorithm of Neumann–Neumann type for solving variational inequalities arising from the elliptic boundary value problems with inequality boundary conditions has been presented. The discretized problem is first turned by the duality theory of convex programming into a quadratic programming problem with bound and equality constraints and the latter is further modified by means of orthogonal projectors to the natural coarse space introduced recently by Farhat and Roux. The resulting problem is then solved by an augmented Lagrangian type algorithm with an outer loop for the Lagrange multipliers for the equality constraints and an inner loop for the solution of the bound constrained quadratic programming problems. The projectors are shown to guarantee an optimal rate of convergence of iterative solution of auxiliary linear problems. Reported theoretical results and numerical experiments indicate high numerical and parallel scalability of the algorithm.

Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique

The paper deals with the numerical solution of the quasi-variational inequality describing the equilibrium of an elastic body in contact with a rigid foundation under Coulomb friction. After a discretization of the problem by mixed finite elements, the duality approach is exploited to reduce the problem to a sequence of quadratic programming problems with box constraints, so that efficient recently proposed algorithms may be applied. A new variant of this method is presented. It combines fixed point with block Gauss-Seidel iterations. The method may be also considered as a new implementation of fixed point iterations for a sequence of problems with given friction. Results of numerical experiments are given showing that the resulting algorithm may be much faster than the original fixed point method and its efficiency is comparable with the solution of frictionless contact problems.

Augmented Lagrangians with Adaptive Precision Control forQuadratic Programming with Equality Constraints

In this paper we introduce an augmented Lagrangian type algorithm for strictly convex quadratic programming problems with equality constraints. The new feature of the proposed algorithm is the adaptive precision control of the solution of auxiliary problems in the inner loop of the basic algorithm. Global convergence and boundedness of the penalty parameter are proved and an error estimate is given that does not have any term that accounts for the inexact solution of the auxiliary problems. Numerical experiments illustrate efficiency of the algorithm presented

An Optimal Algorithm for Minimization of Quadratic Functions with Bounded Spectrum Subject to Separable Convex Inequality and Linear Equality Constraints

An, in a sense, optimal algorithm for minimization of quadratic functions subject to separable convex inequality and linear equality constraints is presented. Its unique feature is an error bound in terms of bounds on the spectrum of the Hessian of the cost function. If applied to a class of problems with the spectrum of the Hessians in a given positive interval, the algorithm can find approximate solutions in a uniformly bounded number of simple iterations, such as matrix-vector multiplications. Moreover, if the class of problems admits a sparse representation of the Hessian, it simply follows that the cost of the solution is proportional to the number of unknowns. Theoretical results are illustrated by numerical experiments.

Cholesky decomposition of a positive semidefinite matrix with known kernel

The Cholesky decomposition of a symmetric positive semidefinite matrix A is a useful tool for solving the related consistent system of linear equations or evaluating the action of a generalized inverse, especially when A is relatively large and sparse. To use the Cholesky decomposition effectively, it is necessary to identify reliably the positions of zero rows or columns of the factors and to choose these positions so that the nonsingular submatrix of A of the maximal rank is reasonably conditioned. The point of this note is to show how to exploit information about the kernel of A to accomplish both tasks. The results are illustrated by numerical experiments.

Duality based domain decomposition with proportioning for the solution of free boundary problems

Quadratic programming problems arising from the discretization of free boundary elliptic problems with the spatial domain comprising several subdomains are considered. An algorithm for the solution of these problems is proposed that combines our algorithm recently presented for the solution of quadratic programming problems with a duality-based domain decomposition method of the Neumann-Neumann type. The characteristic feature of the algorithm is that it reduces the problem to a sequence of well-conditioned auxiliary problems with the precision controlled by the norm of the unbalanced contact residual. The algorithm may be implemented with projections so that it is capable to drop and add many constraints whenever the active set is changed. Numerical experiments with simple model problems indicate that the algorithm is robust and efficient. The algorithm can prove useful in parallel computing environment.

Theoretically supported scalable BETI method for variational inequalities

SummaryThe Boundary Element Tearing and Interconnecting (BETI) methods were recently introduced as boundary element counterparts of the well established Finite Element Tearing and Interconnecting (FETI) methods. Here we combine the BETI method preconditioned by the projector to the “natural coarse grid” with recently proposed optimal algorithms for the solution of bound and equality constrained quadratic programming problems in order to develop a theoretically supported scalable solver for elliptic multidomain boundary variational inequalities such as those describing the equilibrium of a system of bodies in mutual contact. The key observation is that the “natural coarse grid” defines a subspace that contains the solution, so that the preconditioning affects also the non-linear steps. The results are validated by numerical experiments.

Superrelaxation and the rate of convergence in minimizing quadratic functions subject to bound constraints

The paper resolves the problem concerning the rate of convergence of the working set based MPRGP (modified proportioning with reduced gradient projection) algorithm with a long steplength of the reduced projected gradient step. The main results of this paper are the formula for the R-linear rate of convergence of MPRGP in terms of the spectral condition number of the Hessian matrix and the proof of the finite termination property for the problems whose solution does not satisfy the strict complementarity condition. The bound on the R-linear rate of convergence of the projected gradient is also included. For shorter steplengths these results were proved earlier by Dostál and Schöberl. The efficiency of the longer steplength is illustrated by numerical experiments. The result is an important ingredient in developming scalable algorithms for numerical solution of elliptic variational inequalities and substantiates the choice of parameters that turned out to be effective in numerical experiments.