A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints

In this paper we consider optimal control problems subject to a semilinear elliptic state equation together with the control constraints 0≤u≤1 and ∫u=m. Optimality conditions for this problem are derived and reformulated as a nonlinear, nonsmooth equation which is solved using a semismooth Newton method. A regularization of the nonsmooth equation is necessary to obtain the superlinear convergence of the semismooth Newton method. We prove that the solutions of the regularized problems converge to a solution of the original problem and a path-following technique is used to ensure a constant decrease rate of the residual. We show that, in certain situations, the optimal controls take 0–1 values, which amounts to solving a topology optimization problem with volume constraint.

[1]  MICHAEL HINTERMÜLLER,et al.  PDE-Constrained Optimization Subject to Pointwise Constraints on the Control, the State, and Its Derivative , 2009, SIAM J. Optim..

[2]  J. F. Bonnans,et al.  Second-Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems , 1998 .

[3]  Defeng Sun,et al.  A further result on an implicit function theorem for locally Lipschitz functions , 2001, Oper. Res. Lett..

[4]  Michael R. Greenberg,et al.  Chapter 1 – Theory, Methods, and Applications , 1978 .

[5]  Kazufumi Ito,et al.  Novel Concepts for Nonsmooth Optimization and their Impact on Science and Technology , 2011 .

[6]  Fredi Tröltzsch,et al.  Sufficient Second-Order Optimality Conditions for Semilinear Control Problems with Pointwise State Constraints , 2008, SIAM J. Optim..

[7]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[8]  Roland Herzog,et al.  Optimality Conditions and Error Analysis of Semilinear Elliptic Control Problems with L1 Cost Functional , 2012, SIAM J. Optim..

[9]  Grégoire Allaire.  Conception optimale de structures , 2007 .

[10]  Xiaojun Chen,et al.  Smoothing Methods and Semismooth Methods for Nondifferentiable Operator Equations , 2000, SIAM J. Numer. Anal..

[11]  R. Rockafellar,et al.  Implicit Functions and Solution Mappings , 2009 .

[12]  Michal Kočvara,et al.  Nonsmooth approach to optimization problems with equilibrium constraints : theory, applications, and numerical results , 1998 .

[13]  Christian Kanzow,et al.  SC 1 optimization reformulations of the generalized Nash equilibrium problem , 2008, Optim. Methods Softw..

[14]  K. Kunisch,et al.  A duality-based approach to elliptic control problems in non-reflexive Banach spaces , 2011 .

[15]  M. Hintermüller,et al.  Electrical Impedance Tomography: from topology to shape , 2008 .

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

[17]  Eric T. Chung,et al.  Electrical impedance tomography using level set representation and total variational regularization , 2005 .

[18]  Kazufumi Ito,et al.  The Primal-Dual Active Set Method for Nonlinear Optimal Control Problems with Bilateral Constraints , 2004, SIAM J. Control. Optim..

[19]  J. Frédéric Bonnans,et al.  Numerical Optimization: Theoretical and Practical Aspects (Universitext) , 2006 .

[20]  Gerd Wachsmuth,et al.  Convergence and regularization results for optimal control problems with sparsity functional , 2011 .

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

[22]  Martin P. Bendsøe.  Topology Optimization , 2009, Encyclopedia of Optimization.

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

[24]  R. Rockafellar,et al.  Implicit Functions and Solution Mappings: A View from Variational Analysis , 2009 .

[25]  H. Maurer,et al.  On L1‐minimization in optimal control and applications to robotics , 2006 .

[26]  S. D. Chatterji.  Proceedings of the International Congress of Mathematicians , 1995 .

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

[28]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

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

[30]  P. Anselone,et al.  Collectively Compact Operator Approximation Theory and Applications to Integral Equations , 1971 .

[31]  Michael Hinze,et al.  Discrete Concepts in PDE Constrained Optimization , 2009 .

[32]  R. Gorenflo,et al.  Multi-index Mittag-Leffler Functions , 2014 .

[33]  S. Amstutz.  A semismooth Newton method for topology optimization , 2010 .

[34]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[35]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[36]  Georg Stadler,et al.  Elliptic optimal control problems with L1-control cost and applications for the placement of control devices , 2009, Comput. Optim. Appl..

[37]  M. Bendsøe,et al.  Topology Optimization: "Theory, Methods, And Applications" , 2011 .