The semi-smooth Newton method for variationally discretized control constrained elliptic optimal control problems; implementation, convergence and globalization

References

• Bergounioux , M. , Ito , K. and Kunisch , K. 1999 . Primal-dual strategy for constrained optimal control problems . SIAM J. Control Optim. , 37 ( 4 ) : 1176 – 1194 .
   Web of Science ®Google Scholar
• Dziuk , G. 1988 . “ Finite elements for the Beltrami operator on arbitrary surfaces ” . In Partial Differential Equations and Calculus of Variations , Edited by: Stefan , Hildebrandt and Rolf , Leis . 142 – 155 . Berlin/Heidelberg : Springer . Lecture Notes in Mathematics, Vol. 1357
   Google Scholar
• Dziuk , G. and Elliott , C. M. 2007 . Finite elements on evolving surfaces . IMA J. Numer. Anal. , 27 ( 2 ) : 262 – 292 .
   Web of Science ®Google Scholar
• Ekeland , I. and Témam , R. 1999 . “ Convex Analysis and Variational Problems ” . In Unabridged, corrected republication of the 1976 English original, Classics in Applied Mathematics , Vol. 28 , Philadelphia , PA : SIAM .
   Google Scholar
• Gilbarg , D. and Trudinger , N. S. 1998 . “ Elliptic Partial Differential Equations of Second Order ” . Berlin : Springer .
   Google Scholar
• Gräser , C. Globalization of nonsmooth Newton methods for optimal control problems . Günther Of, Olaf Steinbach, Proceedings of ENUMATH 2007, the 7th European conference on numerical mathematics and advanced applications, Graz, Austria . September 10–14 2007 . Numerical Mathematics and Advanced Applications , Edited by: Kunisch , K. pp. 605 – 612 . Berlin : Springer .
   Google Scholar
• Gräser , C. and Kornhuber , R. 2009 . Nonsmooth Newton methods for set-valued saddle point problems . SIAM J. Numer. Anal. , 47 ( 2 ) : 1251 – 1273 .
   Web of Science ®Google Scholar
• Günther , A. and Tber , M. H. 2009 . “ A goal-oriented adaptive Moreau–Yosida algorithm for control- and state-constrained elliptic control problems ” . Preprint No. SPP1253-089, DFG Schwerpunktprogramm 1253
   Google Scholar
• Hintermüller , M. and Ulbrich , M. 2004 . A mesh-independence result for semismooth Newton methods . Math. Program. , 101 : 151 – 184 .
   Web of Science ®Google Scholar
• Hintermüller , M. , Ito , K. and Kunisch , K. 2003 . The primal–dual active set method as a semi-smooth Newton method . SIAM J. Control Optim. , 13 ( 3 ) : 865 – 888 .
   Web of Science ®Google Scholar
• Hinze , M. 2003 . “ A generalized discretization concept for optimal control problems with control constraints ” . Preprint MATH-NM-02-2003, Institut für Numerische Mathematik, Technische Universität Dresden
   Google Scholar
• Hinze , M. 2005 . A variational discretization concept in control constrained optimization: The linear-quadratic case . J. Comput. Optim. Appl. , 30 ( 1 ) : 45 – 61 .
   Web of Science ®Google Scholar
• Hinze , M. and Matthes , U. 2009 . A note on variational discretization of elliptic Neumann boundary control . Control Cybern. , 38 : 577 – 591 .
   Web of Science ®Google Scholar
• Hinze , M. , Pinnau , R. , Ulbrich , M. and Ulbrich , S. 2009 . “ Optimization with PDE constraints. Mathematical Modelling: Theory and Applications, ” . Vol. 23 , Dordrecht : Springer .
   Google Scholar
• Krumbiegel , K. and Rösch , A. 2008 . A new stopping criterion for iterative solvers for control constrained optimal control problems . Arch. Control Sci. , 18 ( 1 ) : 17 – 42 .
   Google Scholar
• Meyer , C. and Rösch , A. 2004 . Superconvergence properties of optimal control problems . SIAM J. Control Optim. , 43 ( 3 ) : 970 – 985 .
   Web of Science ®Google Scholar
• Mignot , F. and Puel , F. P. 1984 . Optimal control in some variational inequalities . SIAM J. Control Optim. , 22 : 466 – 477 .
   Web of Science ®Google Scholar
• Moubachir , M. and Zolésio , J.-P. 2006 . “ Moving Shape Analysis and Control. Applications to Fluid Structure Interactions ” . Boca Raton , FL : Chapman & Hall/CRC .
   Google Scholar
• Rannacher , R. and Scott , R. 1982 . Some optimal error estimates for piecewise linear finite element approximations . Math. Comput. , 38 : 437 – 445 .
   Web of Science ®Google Scholar
• Schiela , A. 2006 . The Control Reduced Interior Point Method. A Function Space Oriented Algorithmic Approach . Hut, München, Verlag Dr. ,
   Google Scholar
• Tröltzsch , F. and Volkwein , S. 2009 . POD a-posteriori error estimates for linear-quadratic optimal control problems . Comput. Optim. Appl. , 44 ( 1 ) : 83 – 115 .
   Web of Science ®Google Scholar
• Ulbrich , M. 2003 . Semismooth Newton methods for operator equations in function spaces . SIAM J. Optim. , 13 : 805 – 841 .
   Web of Science ®Google Scholar
• Ulbrich , M. 2009 . A new mesh-independence result for semismooth Newton methods . Oberwolfach Report , 6 ( 1 ) : 268 – 271 .
   Google Scholar
• Ulbrich , M. 2011 . “ Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces ” . Philadelphia , PA : MOS-SIAM Series on Optimization .
   Google Scholar
• Vierling , M. 2007 . “ Ein semiglattes Newtonverfahren für semidiskretisierte steuerungsbeschränkte Optimalsteuerungsprobleme ” . Diplomarbeit, Universität Hamburg .
   Google Scholar