Structural optimization

, Volume 14, Issue 2–3, pp 100–107 | Cite as

Interior point techniques for optimal control of variational inequalities

  • A. Leôntiev
  • J. Herskovits
Research Papers


This paper is devoted to a new application of an interior point algorithm to solve optimal control problems of variational inequalities. We propose a Lagrangian technique to obtain a necessary optimality system. After the discretization of the optimality system we prove its equivalence to Karush-Kuhn-Tucker conditions of a nonlinear regular minimization problem. This problem can be efficiently solved by using a modification of Herskovits' interior point algorithm for nonlinear optimization. We describe the numerical scheme for solving this problem and give some numerical examples of test problems in 1-D and 2-D.


Civil Engineer Control Problem Variational Inequality Test Problem Minimization Problem 
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  1. Barbu, V. 1984:Optimal control of variational inequalities. Boston: PitmanGoogle Scholar
  2. Barbu, V.; Korman, P. 1991: Approximating optimal control for elliptic obstacle problem by monotone iteration schemes.Numer. Func. Anal. Optimiz. 12, 429–442Google Scholar
  3. Bazaraa, M.S.; Sherali, H.D.; Shetty, C.M. 1993:Nonlinear programming. Theory and algorithms. New York: John WileyGoogle Scholar
  4. Bergounioux, M. 1995: Optimal conditions for optimal control of elliptic problems governed by variational inequalities.Preprint, Université d'Orléans 95-01 Google Scholar
  5. Bermudez, A.; Saguez, C. 1985: Optimal control of variational inequalities.Control & Cybernetics 14, 9–30Google Scholar
  6. Bermudez, A.; Saguez, C. 1987: Optimal control of a Signorini problem.SIAM J. Control & Optimiz. 25, 576–582Google Scholar
  7. Duvaut, G.; Lions, J.-L. 1972:Les inéquations en mécanique et en physique. Paris: DunodGoogle Scholar
  8. Kinderlehrer, D.; Stampacchia, G. 1980:An introduction to variational inequalities and their applications. New York: Academic PressGoogle Scholar
  9. Ekeland, I.; Temam, R. 1976:Convex analysis and variational problems. North-Holland Publishing CompanyGoogle Scholar
  10. Haslinger, J.; Neittaanmäki, P. 1988:Finite element approximation for optimal shape design: theories and applications. New York: John WileyGoogle Scholar
  11. Herskovits, J.N. 1986: A two-stage feasible directions algorithm for nonlinear constrained optimization.Math. Prog. 36, 19–38Google Scholar
  12. Herskovits, J.N. 1992: An interior point technique for nonlinear optimization.Research Report, INRIA, BP 105, 78153 Le Chesnay CEDEX, France,1808 Google Scholar
  13. Herskovits, J.N. 1993: An interior points methods for nonlinearly constrained optimization. In: Rozvany, G.I.N. (ed.)Optimization of large structural systems (Proc. NATO/DFG ASI, held in Berchtesgaden, Germany, 1991), pp. 589–608. Dordrecht: KluwerGoogle Scholar
  14. Leôntiev, A.N. 1994: Existence of a Lagrange multiplier for the control problem of a variational inequality.Control & Cybernetics 23, 809–812Google Scholar
  15. Lions, J.-L. 1969:Quelques méthodes de résolution des problèmes aux limites non linéares. Paris: DunodGoogle Scholar
  16. Luenberger, D.G. 1984:Linear and nonlinear programming (2-nd edition). Addison-WesleyGoogle Scholar
  17. Mignot, F. 1976: Contrôle dans les inéquations variationnelles elliptiques.J. Func. Anal. 22, 130–185Google Scholar
  18. Puel, J.-P. 1987: Some results on optimal control for unilateral problems.Lecture Notes in Control and Information Sciences 114, 225–235Google Scholar
  19. Vautier, I.; Salaun, M.; Herskovits, J.N. 1993: Application of an interior point algorithm to the modeling of unilateral contact between spot-welded shells. In: Herskovits, J.N. (ed.)Structural Optimization '93. COPPE: Rio de JaneiroGoogle Scholar
  20. Zouain, N.A.; Herskovits, J.N.; Borges, L.A.; Feijóo, R.A. 1993: An iterative algorithm for limit analysis with nonlinear yield functions.Int. J. Solids & Struct. 30, 1397–1417Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • A. Leôntiev
    • 1
  • J. Herskovits
    • 1
  1. 1.PEM COPPE/Federal University of Rio de JaneiroRio de JaneiroBrazil

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