521 research outputs found
A class of nonsymmetric preconditioners for saddle point problems
For iterative solution of saddle point problems, a nonsymmetric preconditioning is studied which, with respect to the upper-left block of the system matrix, can be seen as a variant of SSOR. An idealized situation where the SSOR is taken with respect to the skew-symmetric part plus the diagonal part of the upper-left block is analyzed in detail. Since action of the preconditioner involves solution of a Schur complement system, an inexact form of the preconditioner can be of interest. This results in an inner-outer iterative process. Numerical experiments with solution of linearized Navier-Stokes equations demonstrate efficiency of the new preconditioner, especially when the left-upper block is far from symmetric
GMRES-Accelerated ADMM for Quadratic Objectives
We consider the sequence acceleration problem for the alternating direction
method-of-multipliers (ADMM) applied to a class of equality-constrained
problems with strongly convex quadratic objectives, which frequently arise as
the Newton subproblem of interior-point methods. Within this context, the ADMM
update equations are linear, the iterates are confined within a Krylov
subspace, and the General Minimum RESidual (GMRES) algorithm is optimal in its
ability to accelerate convergence. The basic ADMM method solves a
-conditioned problem in iterations. We give
theoretical justification and numerical evidence that the GMRES-accelerated
variant consistently solves the same problem in iterations
for an order-of-magnitude reduction in iterations, despite a worst-case bound
of iterations. The method is shown to be competitive against
standard preconditioned Krylov subspace methods for saddle-point problems. The
method is embedded within SeDuMi, a popular open-source solver for conic
optimization written in MATLAB, and used to solve many large-scale semidefinite
programs with error that decreases like , instead of ,
where is the iteration index.Comment: 31 pages, 7 figures. Accepted for publication in SIAM Journal on
Optimization (SIOPT
Randomized Riemannian Preconditioning for Orthogonality Constrained Problems
Optimization problems with (generalized) orthogonality constraints are
prevalent across science and engineering. For example, in computational science
they arise in the symmetric (generalized) eigenvalue problem, in nonlinear
eigenvalue problems, and in electronic structures computations, to name a few
problems. In statistics and machine learning, they arise, for example, in
canonical correlation analysis and in linear discriminant analysis. In this
article, we consider using randomized preconditioning in the context of
optimization problems with generalized orthogonality constraints. Our proposed
algorithms are based on Riemannian optimization on the generalized Stiefel
manifold equipped with a non-standard preconditioned geometry, which
necessitates development of the geometric components necessary for developing
algorithms based on this approach. Furthermore, we perform asymptotic
convergence analysis of the preconditioned algorithms which help to
characterize the quality of a given preconditioner using second-order
information. Finally, for the problems of canonical correlation analysis and
linear discriminant analysis, we develop randomized preconditioners along with
corresponding bounds on the relevant condition number
Preconditioners for state constrained optimal control problems\ud with Moreau-Yosida penalty function tube
Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the state poses a significant challenge for optimization methods. Our focus here is on the incorporation of the constraints via the Moreau-Yosida regularization technique. This method has been studied recently and has proven to be advantageous compared to other approaches. In this paper we develop preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau-Yosida regularized problem. Numerical results illustrate the competitiveness of this approach. \ud
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Copyright c 2000 John Wiley & Sons, Ltd
Preconditioners for state constrained optimal control problems with Moreau-Yosida penalty function
Optimal control problems with partial differential equations as constraints play an important role in many applications. The inclusion of bound constraints for the state variable poses a significant challenge for optimization methods. Our focus here is on the incorporation of the constraints via the Moreau-Yosida regularization technique. This method has been studied recently and has proven to be advantageous compared to other approaches. In this paper we develop robust preconditioners for the efficient solution of the Newton steps associated with solving the Moreau-Yosida regularized problem. Numerical results illustrate the efficiency of our approach
Hybrid algorithms for cyclically reduced convection-diffusion problems
We consider hybrid and adaptive iterative algorithms for cyclically-reduced discrete convection-diffusion problems. Hybrid algorithms combine via a two phase algorithm, iterative methods which require no a priori information about the coefficient matrix in the first phase with Chebyshev or Richardson iteration in the second phase. For two-dimensional convection-diffusion problems, central difference discretization is considered and the resulting linear system is reduced to approximately half its size by applying one step of cyclic reduction. We examine the numerical performance of the hybrid methods for solving the reduced systems. Our numerical experiments show that for the class of problems considered, an adaptive Chebyshev algorithm that uses modified moments to approximate the eigenvalues requires less work in most cases than the hybrid algorithms based on GMRES/Richardson methods
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