8,074 research outputs found

    GMRES-Accelerated ADMM for Quadratic Objectives

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    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 κ\kappa-conditioned problem in O(κ)O(\sqrt{\kappa}) iterations. We give theoretical justification and numerical evidence that the GMRES-accelerated variant consistently solves the same problem in O(κ1/4)O(\kappa^{1/4}) iterations for an order-of-magnitude reduction in iterations, despite a worst-case bound of O(κ)O(\sqrt{\kappa}) 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 O(1/k2)O(1/k^{2}), instead of O(1/k)O(1/k), where kk is the iteration index.Comment: 31 pages, 7 figures. Accepted for publication in SIAM Journal on Optimization (SIOPT

    Preconditioners for state constrained optimal control problems with Moreau-Yosida penalty function

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    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

    All-at-once preconditioning in PDE-constrained optimization

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    The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDE-constrained optimization and show how the all-at-once approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound constraints for the control are introduced. Numerical results will illustrate the competitiveness of our techniques
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