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Duality Theory and Applications to Unilateral Problems

Journal of Optimization Theory and Applications volume 162, pages 718–734 (2014)Cite this article

Abstract

This paper is concerned with the problem of strong duality between an infinite dimensional convex optimization problem with cone and equality constraints and its Lagrange dual.

A necessary and sufficient condition and sufficient conditions, really new, in order that the strong duality holds true are given. As an application, the existence of the Lagrange multiplier associated with the obstacle problem and to an elastic–plastic torsion problem, more general than the ones previously considered, is stated together with a characterization of the elastic–plastic torsion problem. This application is the main result of the paper.

It is worth remarking that the usual conditions based on the interior, on the core, on the intrinsic core or on the strong quasi-relative interior cannot be used because they require the nonemptiness of the interior (and of the above mentioned generalized interior concepts) of the ordering cone, which is usually empty.

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References

  1. Rockafellar, R.T.: Conjugate Duality and Optimization. Conference board of the Mathematical Science regional Conference Series in Applied Mathematics, vol. 16. Society for Industrial and Applied Mathematics, Philadelphia (1974)

    Book  MATH  Google Scholar 

  2. Holmes, R.B.: Geometric Functional Analysis. Springer, Berlin (1975)

    MATH  Google Scholar 

  3. Borwein, J.M., Jeyakumar, V., Lewis, A.S., Wolkowicz, M.: Constrained approximation via convex programming. Preprint, University of Waterloo (1988)

  4. Jeyakumar, V., Wolkowicz, H.: Generalizations of slater constraint qualification for infinite convex programs. Math. Program. 57, 85–101 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barbagallo, A., Maugeri, A.: Duality theory for a dynamic oligopolistic market equilibrium problem. Optimization 60, 29–52 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cojocaru, M.G., Daniele, P., Nagurney, A.: Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces and applications. J. Optim. Theory Appl. 127, 549–563 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Daniele, P.: Dynamic Networks and Evolutionary Variational Inequalities. Edward Elgar, Cheltenham (2006)

    MATH  Google Scholar 

  8. Daniele, P.: Evolutionary variational inequalities and applications to complex dynamic multi-level models. Transportation Research Part E 46, 855–880 (2010)

    Article  Google Scholar 

  9. Daniele, P., Giuffré, S.: General infinite dimensional duality and applications to evolutionary network equilibrium problems. Optim. Lett. 1, 227–243 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Daniele, P., Giuffré, S., Idone, G., Maugeri, A.: Infinite dimensional duality and applications. Math. Ann. 339, 221–239 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Donato, M.B., Maugeri, A., Milasi, M., Vitanza, C.: Duality theory for a dynamic Walrasian pure exchange economy. Pac. J. Optim. 4, 537–547 (2008)

    MATH  MathSciNet  Google Scholar 

  12. Gwinner, J., Raciti, F.: Random equilibrium problems on networks. Math. Comput. Model. 43, 880–891 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gwinner, J., Raciti, F.: On a class of random variational inequalities on random sets. Numer. Funct. Anal. Optim. 27, 619–636 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Idone, G., Maugeri, A.: Generalized constraints qualification and infinite dimensional duality. Taiwan. J. Math. 13, 1711–1722 (2009)

    MATH  MathSciNet  Google Scholar 

  15. Scrimali, L.: Infinite dimensional duality theory applied to investment strategies in environmental policy. J. Optim. Theory Appl. 154, 258–277 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Brezis, H.: Moltiplicateur de Lagrange en torsion elasto-plastique. Arch. Ration. Mech. Anal. 49, 32–40 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bot, R.I., Csetnek, E.R., Moldovan, A.: Revisiting some dual theorems via the quasirelative interior in convex optimization. J. Optim. Theory Appl. 139, 67–84 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Daniele, P., Giuffré, S., Maugeri, A.: Remarks on general infinite dimensional duality with cone and equality constraints. Commun. Appl. Anal. 13(4), 567–578 (2009)

    MATH  MathSciNet  Google Scholar 

  19. Grad, A.: Quasi-relative interior-type constraints qualifications ensuring strong Lagrange duality for optimization problems with cone and affine constraints. J. Math. Anal. Appl. 364, 86–95 (2010)

    Article  MathSciNet  Google Scholar 

  20. Maugeri, A., Raciti, F.: Remarks on infinite dimensional duality. J. Glob. Optim. 46, 581–588 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Zalinescu, C.: Private communications: September 4th, 5th, 7th, October 10th (2007)

  22. Brezis, H., Sibony, M.: Equivalence de Deux Inéquation Variationnelles et Application. Arch. Ration. Mech. Anal. 41, 254–265 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lanchon, H., Duvaut, G.: Sur la Solution du probleme de torsion élastoplastic d’une bare cylindrique de section quelconque. C. R. Acad. Sci. Paris, Ser. A 264, 520–523 (1967)

    MATH  MathSciNet  Google Scholar 

  24. Lanchon, H.: Solution du probleme de torsion élastoplastic d’une bare cylindrique de section quelconque. C. R. Acad. Sci. Paris, Ser. A 269, 791–794 (1969)

    MATH  MathSciNet  Google Scholar 

  25. Rodriguez, J.F.: Obstacle Problems in Mathematical Physics. North-Holland, Amsterdam (1987)

    Google Scholar 

  26. Bot, R.I., Csetnek, E.R., Wanka, G.: Regularity conditions via quasi-relative interior in convex programming. SIAM J. Optim. 19(1), 217–233 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Borwein, J.M., Lewis, A.S.: Partially finite convex programming, part I: quasi relative interiors and duality theory. Math. Program. 57, 15–48 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  28. Daniele, P., Maugeri, A.: Variational inequalities and discrete continuum models of network equilibrium. Probl. Math. Comput. Model. 35, 689–708 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  29. Daniele, P., Giuffré, S., Pia, S.: Competitive financial equilibrium problems with policy interventions. J. Ind. Manag. Optim. 1(1), 39–52 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Donato, M.B., Milasi, M., Vitanza, C.: Quasi-variational approach of a competitive economic equilibrium problem with utility function: existence of equilibrium. Math. Models Methods Appl. Sci. 18(3), 351–367 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Giannessi, F., Maugeri, A.: Preface of the special issue on the proceedings of the first AMS-UMI joint meeting. J. Glob. Optim. 28 (2004)

  32. Idone, G., Maugeri, A.: Variational inequalities and transport planning for an elastic and continuous model. J. Ind. Manag. Optim. 1, 81–86 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  33. Maugeri, A., Murthy, M.K.V., Trudinger, N.: Preface of the variational analysis and partial differential equations. In memory of Sergio Campanato and Guido Stampacchia. J. Glob. Optim. 40, 1–5 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  34. Maugeri, A., Raciti, F.: On general infinite dimensional complementarity problems. Optim. Lett. 2, 71–90 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  35. Prager, W., Hodge, P.G.: Theory of Perfectly Plastic Solids. Wiley, New York (1951)

    MATH  Google Scholar 

  36. Ting, T.W.: Elastic-plastic torsion of a square bar. Trans. Am. Math. Soc. 113, 369–401 (1966)

    MathSciNet  Google Scholar 

  37. Schultze, R.S.R.: Von Mises (1883–1953): a pioneer of applied mathematics in four countries. Newsl.—Eur. Math. Soc. 73, 31–34 (2009)

    MATH  Google Scholar 

  38. Von Mises, R.: Mechanik der festen Körper im plastish deformablem Zustand. Nachr. Ges. Wiss. Goett., Math.-Phys. Kl. 582–592, (1913)

  39. Von Mises, R.: Three Remarks on the Theory of the Ideal Plastic Body. Reissner Anniversary Volume. Edwards, Ann Arbor (1949)

    Google Scholar 

  40. Ting, T.W.: Elastic-plastic torsion problem II. Arch. Ration. Mech. Anal. 25, 342–366 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  41. Ting, T.W.: Elastic-plastic torsion problem III. Arch. Ration. Mech. Anal. 34, 228–244 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  42. Friedman, A.: Variational Principles and Free Boundary Problems. Wiley, New York (1982)

    MATH  Google Scholar 

  43. Brezis, H.: Problémes Unilatéraux. J. Math. Pures Appl. 51, 1–168 (1972)

    MATH  MathSciNet  Google Scholar 

  44. Stampacchia, G.: On a problem of numerical analysis connected with the theory of variational inequalities. In: Symposia Mathematica, vol. X, pp. 281–293. Academic Press, San Diego (1972)

    Google Scholar 

  45. Chiadó-Piat, V., Percivale, D.: Generalized Lagrange multipliers in elastoplastic torsion. J. Differ. Equ. 114, 570–579 (1994)

    Article  MATH  Google Scholar 

  46. Donato, M.B.: The infinite dimensional Lagrange multiplier rule for convex optimization problems. J. Funct. Anal. 261(8), 2083–2093 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  47. Giuffrè, S., Maugeri, A.: New results on infinite dimensional duality in elastic–plastic torsion. Filomat 26, 1029–1036 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6–I, 95125, Catania, Italy

    Patrizia Daniele, Antonino Maugeri & Fabio Raciti

  2. D.I.I.E.S, Mediterranean University of Reggio Calabria, Loc. Feo di Vito, 89060, Reggío Calabria, Italy

    Sofia Giuffrè

Authors
  1. Patrizia Daniele
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  2. Sofia Giuffrè
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  3. Antonino Maugeri
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  4. Fabio Raciti
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Correspondence to Sofia Giuffrè.

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Daniele, P., Giuffrè, S., Maugeri, A. et al. Duality Theory and Applications to Unilateral Problems. J Optim Theory Appl 162, 718–734 (2014). https://doi.org/10.1007/s10957-013-0512-4

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  • DOI: https://doi.org/10.1007/s10957-013-0512-4

Keywords

  • Strong duality
  • Variational inequalities
  • Quasi-relative interior
  • Elastic–plastic torsion