Abstract
In the present work we apply an augmented Lagrange method to solve pointwise state constrained elliptic optimal control problems. We prove strong convergence of the primal variables as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. In addition, we show that the sequence of generated penalty parameters is bounded only in exceptional situations, which is different from classical results in finite-dimensional optimization. In addition, numerical results are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bergounioux, M.: Augmented Lagrangian method for distributed optimal control problems with state constraints. J. Optim. Theory Appl. 78(3), 493–521 (1993)
Bergounioux, M.: On boundary state constrained control problems. Numer. Funct. Anal. Optim. 14(5–6), 515–543 (1993)
Bergounioux, M., Kunisch, K.: Augmented Lagrangian techniques for elliptic state constrained optimal control problems. SIAM J. Control Optim. 35(5), 1524–1543 (1997)
Birgin, E.G., Castillo, R.A., Martínez, J.M.: Numerical comparison of augmented Lagrangian algorithms for nonconvex problems. Comput. Optim. Appl. 31(1), 31–55 (2005)
Birgin, E.G., Martínez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2014)
Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24(6), 1309–1318 (1986)
Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31(4), 993–1006 (1993)
De los Reyes, J.C.: Numerical PDE-Constrained Optimization. Springer, Heidelberg (2015)
Hintermüller, M., Hinze, M.: Moreau–Yosida regularization in state constrained elliptic control problems: error estimates and parameter adjustment. SIAM J. Numer. Anal. 47(3), 1666–1683 (2009)
Hintermüller, M., Kunisch, K.: Feasible and noninterior path-following in constrained minimization with low multiplier regularity. SIAM J. Control Optim. 45(4), 1198–1221 (2006)
Hintermüller, M., Kunisch, K.: Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17(1), 159–187 (2006). (electronic)
Hintermüller, M., Kunisch, K.: PDE-constrained optimization subject to pointwise constraints on the control, the state, and its derivative. SIAM J. Optim. 20(3), 1133–1156 (2009)
Hintermüller, M., Schiela, A., Wollner, W.: The length of the primal-dual path in Moreau–Yosida-based path-following methods for state constrained optimal control. SIAM J. Optim. 24(1), 108–126 (2014)
Hinze, M., Meyer, C.: Variational discretization of Lavrentiev-regularized state constrained elliptic optimal control problems. Comput. Optim. Appl. 46(3), 487–510 (2010)
Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints, Volume 23 of Mathematical Modelling: Theory and Applications. Springer, New York (2009)
Ito, K., Kunisch, K.: Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. 41, 591–616 (2000). (5-6, Ser. A: Theory Methods)
Ito, K., Kunisch, K.: Semi-smooth Newton methods for state-constrained optimal control problems. Syst. Control Lett. 50(3), 221–228 (2003)
Ito, K., Kunisch, K.: Semi-smooth Newton methods for variational inequalities of the first kind. M2AN Math. Model. Numer. Anal. 37(1), 41–62 (2003)
Kanzow, C., Steck, D., Wachsmuth D.: An augmented Lagrangian method for optimization problems in Banach spaces. Preprint SPP1962-003 of priority program “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization” (SPP 1962) (2016)
Krumbiegel, K., Neitzel, I., Rösch, A.: Sufficient optimality conditions for the Moreau–Yosida-type regularization concept applied to semilinear elliptic optimal control problems with pointwise state constraints. Ann. Acad. Rom. Sci. Ser. Math. Appl. 2(2), 222–246 (2010)
Krumbiegel, K., Neitzel, I., Rösch, A.: Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints. Comput. Optim. Appl. 52(1), 181–207 (2012)
Kruse, F., Ulbrich, M.: A self-concordant interior point approach for optimal control with state constraints. SIAM J. Optim. 25(2), 770–806 (2015)
Logg, A., Mardal, K.-A., Wells, G.N., et al.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)
Logg, A., Wells, G.N.: DOLFIN: automated finite element computing. ACM Trans. Math. Softw. 37(2), 20 (2010)
Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33(2–3), 209–228 (2006)
Rösch, A., Wachsmuth, D.: A-posteriori error estimates for optimal control problems with state and control constraints. Numer. Math. 120(4), 733–762 (2012)
Schiela, A.: An interior point method in function space for the efficient solution of state constrained optimal control problems. Math. Program. 138(1–2, Ser. A), 83–114 (2013)
Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the German Research Foundation (DFG) within the priority program “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization” (SPP 1962) under Grant Number Wa 3626/3-1.
Rights and permissions
About this article
Cite this article
Karl, V., Wachsmuth, D. An augmented Lagrange method for elliptic state constrained optimal control problems. Comput Optim Appl 69, 857–880 (2018). https://doi.org/10.1007/s10589-017-9965-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-017-9965-y