Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T08:37:51.186Z Has data issue: false hasContentIssue false
  • English
  • Français

A FEM-Multigrid Scheme for Elliptic Nash-Equilibrium Multiobjective Optimal Control Problems

Published online by Cambridge University Press:  28 May 2015

Mohammad Tanvir Rahman  and
Alfio Borzì
Mohammad Tanvir Rahman
Affiliation:
Department of Mathematics, University of Würzburg, Emil-Fischer-Str. 30, 97074 Würzburg, Germany
Alfio Borzì*
Affiliation:
Department of Mathematics, University of Würzburg, Emil-Fischer-Str. 30, 97074 Würzburg, Germany
*
*Email addresses: [email protected] (Mohammad Tanvir Rahman), [email protected], (Alfio Borzí)
Get access

Abstract

A finite-element multigrid scheme for elliptic Nash-equilibrium multiobjective optimal control problems with control constraints is investigated. The multigrid computational framework implements a nonlinear multigrid strategy with collective smoothing for solving the multiobjective optimality system discretized with finite elements. Error estimates for the optimal solution and two-grid local Fourier analysis of the multigrid scheme are presented. Results of numerical experiments are presented to demonstrate the effectiveness of the proposed framework.

Keywords

15A1265F1065F15Elliptic partial differential equationsmultigrid methodsNash equilibriumoptimal control theoryfinite element method
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 2 , May 2015 , pp. 253 - 282
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Borzì, A. and Kanzow, C., Formulation and numerical solution of nash equilibrium multiobjective elliptic control problems, SIAM J. Control Optim., 51(2013), 718–744.CrossRefGoogle Scholar
[2]Borzì, A. and Schulz, V., Computational optimization of systems governed by partial differential equations, SIAM book series on Computational Science and Engineering 08, SIAM, Philadelphia, PA, 2012.Google Scholar
[3]Borzì, A. and Kunisch, K., A multigrid scheme for elliptic constrained optimal control problems, Comput. Optim. Appl., 31 (2005), pp. 309–333.CrossRefGoogle Scholar
[4]Brandt, A., Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, GMD Studies 85, GMD-FTT, 1985.Google Scholar
[5]Trottenberg, U. and Schuller, A., Multigrid, Academic Press, Inc., Orlando, FL, 2001.Google Scholar
[6]Gockenbach, M.S., Understanding and implementing the finite element method, SIAM 2006.CrossRefGoogle Scholar
[7]Chinchuluun, A., Pardalos, P.M., Migdalas, A., and Pitsoulis, L. (Eds.), Pareto Optimality, Game Theory and Equilibria, Springer, 2008.CrossRefGoogle Scholar
[8]Dempe, S., Foundations ofBilevel Programming, Kluwer Academic Press, Dordrecht, 2002.Google Scholar
[9]Désidéri, J.-A., Cooperation and competition in multidisciplinary optimization. Application to the aero-structural aircraft wing shape optimization. Comput. Optim. Appl., 52 (2012), pp. 29–68.CrossRefGoogle Scholar
[10]Ehrgott, M., Multicriteria Optimization, Springer, 2nd edition 2005.Google Scholar
[11]Gauger, N., Ilic, C., Schmidt, S. and Schulz, V., Non-parametric aerodynamic shape optimization, in Leugering, G., Engell, S., Griewank, A., Hinze, M., Rannacher, R., Schulz, V., Ulbrich, M., and Ulbrich, S. (eds.), Constrained Optimization and Optimal Control for Partial Differential Equations, International Series of Numerical Mathematics, vol, 160, pp. 289–300, Birkhäuser 2012.CrossRefGoogle Scholar
[12]Lions, J.L., Controle de Pareto de systemes distribues: Le cas devolution. Comptes Rendus de l’Academie des Sciences, Serie I, 302 (1986), pp. 413–417.Google Scholar
[13]Liu, G.P., Yang, J.B., and Whidborne, J.F., Multiobjective Optimisation and Control, Research Studies Press LTD, 2001.Google Scholar
[14]Ramos, A.M., Glowinski, R., and Periaux, J., Nash equilibria for the multiobjective control of linear partial differential equations, J. Optim. Theory Appl., 112 (2002), pp. 457–498.CrossRefGoogle Scholar
[15]Ramos, A.M., Glowinski, R., and Periaux, J., Pointwise control of the Burgers equation and related Nash equilibrium problems: Computational approach., J. Optim. Theory Appl., 112 (2002), pp. 499–516.CrossRefGoogle Scholar
[16]Rösch, A., Error estimates for linear-quadratic control problems with control constraints, Optimization Methods and Software 21 (2006), 121–134.CrossRefGoogle Scholar
[17]Stadler, W., Fundamentals of multicriteria optimization, In Multicriteria optimization in engineering and in the sciences, volume 37 of Math. Concepts Methods Sci. Engrg., pages 1–25. Plenum, New York, 1988.Google Scholar
[18]Hintermüller, M. and Surowiec, T., APDE-constrained generalized Nash equilibrium problem with pointwise control and state constraints., Pacific J. on Optimization 9(2), 251–273, 2013Google Scholar