Preconditioning for partial differential equation constrained optimization with control constraints

SUMMARY Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the control poses a significant additional challenge for optimization methods. In this paper, we propose preconditioners for the saddle point problems that arise when a primal–dual active set method is used. We also show for this method that the same saddle point system can be derived when the method is considered as a semismooth Newton method. In addition, the projected gradient method can be employed to solve optimization problems with simple bounds, and we discuss the efficient solution of the linear systems in question. In the case when an acceleration technique is employed for the projected gradient method, this again yields a semismooth Newton method that is equivalent to the primal–dual active set method. We also consider the Moreau–Yosida regularization method for control constraints and efficient preconditioners for this technique. Numerical results illustrate the competitiveness of these approaches. Copyright © 2011 John Wiley & Sons, Ltd.

[1]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[2]  T. Rees,et al.  Block‐triangular preconditioners for PDE‐constrained optimization , 2010, Numer. Linear Algebra Appl..

[3]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[4]  Kazufumi Ito,et al.  Semi-smooth Newton methods for state-constrained optimal control problems , 2003, Syst. Control. Lett..

[5]  Carl Tim Kelley,et al.  Iterative methods for optimization , 1999, Frontiers in applied mathematics.

[6]  Martin Stoll,et al.  Combination Preconditioning and the Bramble-Pasciak+ Preconditioner , 2008, SIAM J. Matrix Anal. Appl..

[7]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[8]  A. Wathen,et al.  Chebyshev semi-iteration in preconditioning for problems including the mass matrix. , 2008 .

[9]  J. Pasciak,et al.  A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems , 1988 .

[10]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[11]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[12]  Michael Griebel,et al.  A multigrid method for constrained optimal control problems , 2011, J. Comput. Appl. Math..

[13]  K. Kunisch,et al.  Primal-Dual Strategy for Constrained Optimal Control Problems , 1999 .

[14]  H. Elman.  Multigrid and Krylov subspace methods for the discrete Stokes equations , 1994 .

[15]  Kazufumi Ito,et al.  Lagrange multiplier approach to variational problems and applications , 2008, Advances in design and control.

[16]  T. Manteuffel,et al.  A taxonomy for conjugate gradient methods , 1990 .

[17]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[18]  Liqun Qi,et al.  Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations , 1993, Math. Oper. Res..

[19]  Liqun Qi,et al.  A nonsmooth version of Newton's method , 1993, Math. Program..

[20]  José Mario Martínez,et al.  Large-Scale Active-Set Box-Constrained Optimization Method with Spectral Projected Gradients , 2002, Comput. Optim. Appl..

[21]  Kazufumi Ito,et al.  The Primal-Dual Active Set Strategy as a Semismooth Newton Method , 2002, SIAM J. Optim..

[22]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .

[23]  Roger Fletcher,et al.  Practical methods of optimization; (2nd ed.) , 1987 .

[24]  F. Tröltzsch.  Optimale Steuerung partieller Differentialgleichungen , 2005 .

[25]  Ivan P. Gavrilyuk,et al.  Lagrange multiplier approach to variational problems and applications , 2010, Math. Comput..

[26]  Paul E. Saylor,et al.  The Role of the Inner Product in Stopping Criteria for Conjugate Gradient Iterations , 2001 .

[27]  Valeria Simoncini,et al.  Block triangular preconditioners for symmetric saddle-point problems , 2004 .

[28]  Philip E. Gill,et al.  Practical optimization , 1981 .

[29]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .

[30]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[31]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[32]  A. Wathen.  Realistic Eigenvalue Bounds for the Galerkin Mass Matrix , 1987 .

[33]  Martin Stoll,et al.  All-at-once preconditioning in PDE-constrained optimization , 2010, Kybernetika.

[34]  D. Bertsekas.  Projected Newton methods for optimization problems with simple constraints , 1981, CDC 1981.

[35]  Stefan Ulbrich,et al.  Optimization with PDE Constraints , 2008, Mathematical modelling.

[36]  Andrew J. Wathen,et al.  Optimal Solvers for PDE-Constrained Optimization , 2010, SIAM J. Sci. Comput..

[37]  R. Varga,et al.  Chebyshev semi-iterative methods, successive overrelaxation iterative methods, and second order Richardson iterative methods , 1961 .

[38]  W. Bangerth,et al.  deal.II—A general-purpose object-oriented finite element library , 2007, TOMS.

[39]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[40]  Martin Stoll,et al.  Preconditioners for state‐constrained optimal control problems with Moreau–Yosida penalty function , 2014, Numer. Linear Algebra Appl..

[41]  Mark Frederick Hoemmen,et al.  An Overview of Trilinos , 2003 .

[42]  採編典藏組.  Society for Industrial and Applied Mathematics(SIAM) , 2008 .

[43]  Ekkehard W. Sachs,et al.  Preconditioned Conjugate Gradient Method for Optimal Control Problems with Control and State Constraints , 2010, SIAM J. Matrix Anal. Appl..