Nonlinear Duality in Banach Spaces and Applications to Finance and Elasticity

  • G. Colajanni
  • Patrizia DanieleEmail author
  • Sofia Giuffrè
  • Antonino Maugeri
Part of the Springer Optimization and Its Applications book series (SOIA, volume 134)


In this chapter we first present some theoretic concepts related to the strong duality in the infinite-dimensional setting. Then, we apply such results to the general financial equilibrium economy, studying also the dual formulation of the problem, analyzing both the sector’s and the system’s viewpoints and deriving the contagion phenomenon. Further, we provide an evolutionary Markowitz-type measure of the risk with a memory term. Finally, we apply Assumption S to the elastic-plastic torsion problem for linear operators and investigate the existence of Lagrange multipliers to the elastic-plastic torsion problem for nonlinear monotone operators, providing an example of the so-called Von Mises functions and searching for radial solutions.


  1. 1.
    A. Barbagallo, A. Maugeri, Duality theory for a dynamic oligopolistic market equilibrium problem. Optimization 60, 29–52 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Barbagallo, P. Daniele, S. Giuffrè, A. Maugeri, Variational approach for a general financial equilibrium problem: the deficit formula, the balance law and the liability formula. A path to the economy recovery. Eur. J. Oper. Res. 237(1), 231–244 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    J.M. Borwein, V. Jeyakumar, A.S. Lewis, M. Wolkowicz, Constrained approximation via convex programming. University of Waterloo. Preprint (1988)Google Scholar
  4. 4.
    R.I. Bot, E.R. Csetnek, A. Moldovan, Revisiting some duality theorems via the quasirelative interior in convex optimization. J. Optim. Theory Appl. 139(1), 67–84 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    H. Brezis, Moltiplicateur de Lagrange en Torsion Elasto-Plastique. Arch. Rational Mech. Anal. 49, 32–40 (1972)MathSciNetCrossRefGoogle Scholar
  6. 6.
    H. Brezis, Problèmes Unilatéraux. J. Math. Pures Appl. 51, 1–168 (1972)Google Scholar
  7. 7.
    H. Brezis, G. Stampacchia, Sur la régularité de la solution d’inéquations elliptiques. Bull. Soc. Math. Fr. 96, 153–180 (1968)CrossRefGoogle Scholar
  8. 8.
    V. Caruso, P. Daniele, A network model for minimizing the total organ transplant costs. Eur. J. Oper. Res. (2017).
  9. 9.
    V. Chiadó-Piat, D. Percivale, Generalized Lagrange multipliers in elastoplastic torsion, J. Differ. Equ. 114, 570–579 (1994)MathSciNetCrossRefGoogle Scholar
  10. 10.
    M.G. Cojocaru, P. Daniele, A. Nagurney, Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces and applications. J. Optim. Theory Appl. 127, 549–563 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    P. Daniele, Dynamic Networks and Evolutionary Variational Inequalities (Edward Elgar Publishing, Cheltenham, 2006)zbMATHGoogle Scholar
  12. 12.
    P. Daniele, Evolutionary variational inequalities and applications to complex dynamic multi-level models. Transp. Res. Part E 46, 855–880 (2010)CrossRefGoogle Scholar
  13. 13.
    P. Daniele, S. Giuffrè, General infinite dimensional duality and applications to evolutionary network equilibrium problems. Optim. Lett. 1, 227–243 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    P. Daniele, S. Giuffrè, Random variational inequalities and the random traffic equilibrium problem. J. Optim. Theory Appl. 167(1), 363–381 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    P. Daniele, S. Giuffrè, S. Pia, Competitive financial equilibrium problems with policy interventions. J. Ind. Manag. Optim. 1(1), 39–52 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    P. Daniele, S. Giuffrè, G. Idone, A. Maugeri, Infinite dimensional duality and applications. Math. Ann. 339, 221–239 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    P. Daniele, S. Giuffrè, A. Maugeri, Remarks on general infinite dimensional duality with cone and equality constraints. Commun. Appl. Anal. 13(4), 567–578 (2009)MathSciNetzbMATHGoogle Scholar
  18. 18.
    P. Daniele, S. Giuffrè, M. Lorino, A. Maugeri, C. Mirabella, Functional inequalities and analysis of contagion in the financial networks, in Handbook of Functional Equations – Functional Inequalities, ed. by Th.M. Rassias. Optimization and Its Applications, vol. 95 (Springer, Berlin, 2014), pp. 129–146zbMATHGoogle Scholar
  19. 19.
    P. Daniele, S. Giuffrè, A. Maugeri, F. Raciti, Duality theory and applications to unilateral problems. J. Optim. Theory Appl. 162(3), 718–734 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    P. Daniele, S. Giuffrè, M. Lorino, Functional inequalities, regularity and computation of the deficit and surplus variables in the financial equilibrium problem. J. Glob. Optim. 65, 575–596 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    P. Daniele, M. Lorino, C. Mirabella, The financial equilibrium problem with a Markowitz-type memory term and adaptive, constraints. J. Optim. Theory Appl. 171, 276–296 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    P. Daniele, A. Maugeri, A. Nagurney, Cybersecurity investments with nonlinear budget constraints: analysis of the marginal expected utilities, in Operations Research, Engineering, and Cyber Security, ed. by N.J. Daras, T.M. Rassias. Springer Optimization and Its Applications, vol. 113 (Springer, Berlin, 2017), pp. 117–134Google Scholar
  23. 23.
    M.B. Donato, The infinite dimensional Lagrange multiplier rule for convex optimization problems. J. Funct. Anal. 261(8), 2083–2093 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    M.B. Donato, A. Maugeri, M. Milasi, C. Vitanza, Duality theory for a dynamic Walrasian pure exchange economy. Pac. J. Optim. 4, 537–547 (2008)MathSciNetzbMATHGoogle Scholar
  25. 25.
    S. Giuffrè, Strong solvability of boundary value contact problems. Appl. Math. Optim. 51(3), 361–372 (2005)MathSciNetCrossRefGoogle Scholar
  26. 26.
    S. Giuffrè, Elements of duality theory, in Topics in Nonlinear Analysis and Optimization, ed. by Q.H. Ansari (World Education, Delhi, 2012), pp. 251–267Google Scholar
  27. 27.
    S. Giuffrè, S. Pia, Weighted traffic equilibrium problem in non pivot Hilbert spaces with long term memory, in AIP Conference Proceedings Rodi, September 2010, vol. 1281, pp. 282–285Google Scholar
  28. 28.
    S. Giuffrè, A. Maugeri, New results on infinite dimensional duality in elastic-plastic torsion. Filomat 26(5), 1029–1036 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    S. Giuffrè, A. Maugeri, Lagrange multipliers in elastic-plastic torsion, in AIP Conference Proceedings Rodi, September 2013, vol. 1558, pp. 1801–1804Google Scholar
  30. 30.
    S. Giuffrè, A. Maugeri, A measure-type Lagrange multiplier for the elastic-plastic torsion. Nonlinear Anal. 102, 23–29 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    S. Giuffrè, G. Idone, A. Maugeri, Duality theory and optimality conditions for generalized complementary problems. Nonlinear Anal. 63, e1655–e1664 (2005)CrossRefGoogle Scholar
  32. 32.
    S. Giuffrè, A. Maugeri, D. Puglisi, Lagrange multipliers in elastic-plastic torsion problem for nonlinear monotone operators. J. Differ. Equ. 259(3), 817–837 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    S. Giuffrè, A. Pratelli, D. Puglisi, Radial solutions and free boundary of the elastic-plastic torsion problem. J. Convex Anal. 25(2), 529–543 (2018)MathSciNetzbMATHGoogle Scholar
  34. 34.
    J. Gwinner, F. Raciti, Random equilibrium problems on networks. Math. Comput. Model. 43, 880–891 (2006)MathSciNetCrossRefGoogle Scholar
  35. 35.
    J. Gwinner, F. Raciti, On a class of random variational inequalities on random sets. Numer. Funct. Anal. Optim. 27, 619–636 (2006)MathSciNetCrossRefGoogle Scholar
  36. 36.
    R.B. Holmes, Geometric Functional Analysis (Springer, Berlin, 1975)zbMATHGoogle Scholar
  37. 37.
    G. Idone, A. Maugeri, Generalized constraints qualification and infinite dimensional duality. Taiwan. J. Math. 13, 1711–1722 (2009)MathSciNetCrossRefGoogle Scholar
  38. 38.
    J. Jahn, Introduction to the Theory of Nonlinear Optimization, 3rd edn. (Springer, Berlin, 2007)zbMATHGoogle Scholar
  39. 39.
    V. Jeyakumar, H. Wolkowicz, Generalizations of slater constraint qualification for infinite convex programs. Math. Program. 57, 85–101 (1992)MathSciNetCrossRefGoogle Scholar
  40. 40.
    H.M. Markowitz, Portfolio selection. J. Financ. 7, 77–91 (1952)Google Scholar
  41. 41.
    H.M. Markowitz, Portfolio Selection: Efficient Diversification of Investments (Wiley, New York, 1959)Google Scholar
  42. 42.
    A. Maugeri, D. Puglisi, A new necessary and sufficient condition for the strong duality and the infinite dimensional Lagrange Multiplier rule. J. Math. Anal. Appl. 415(2), 661–676 (2014)MathSciNetCrossRefGoogle Scholar
  43. 43.
    A. Maugeri, D. Puglisi, Non-convex strong duality via subdifferential. Numer. Funct. Anal. Optim. 35, 1095–1112 (2014)MathSciNetCrossRefGoogle Scholar
  44. 44.
    A. Maugeri, D. Puglisi, On nonlinear strong duality and the infinite dimensional Lagrange multiplier rule. J. Nonlinear Convex Anal. 18(3), 369–378 (2017)MathSciNetzbMATHGoogle Scholar
  45. 45.
    A. Maugeri, F. Raciti, Remarks on infinite dimensional duality. J. Glob. Optim. 46, 581–588 (2010)MathSciNetCrossRefGoogle Scholar
  46. 46.
    A. Maugeri, L. Scrimali, New approach to solve convex infinite-dimensional bilevel problems: application to the pollution emission price problem. J. Optim. Theory Appl. 169(2), 370–387 (2016)MathSciNetCrossRefGoogle Scholar
  47. 47.
    R.T. Rockafellar, Conjugate duality and optimization, in Conference Board of the Mathematical Science Regional Conference Series in Applied Mathematics, vol. 16 (Society for Industrial and Applied Mathematics, Philadelphia, 1974)Google Scholar
  48. 48.
    J.F. Rodrigues, Obstacle Problems in Mathematical Physics. Mathematics Studies, vol. 134 (Elsevier, Amsterdam, 1987)CrossRefGoogle Scholar
  49. 49.
    L. Scrimali, Infinite dimensional duality theory applied to investment strategies in environmental policy. J. Optim. Theory Appl. 154, 258–277 (2012)MathSciNetCrossRefGoogle Scholar
  50. 50.
    T.W. Ting, Elastic-plastic torsion of a square bar. Trans. Am. Math. Soc. 113, 369–401 (1966)zbMATHGoogle Scholar
  51. 51.
    T.W. Ting, Elastic-plastic torsion problem II. Arch. Ration. Mech. Anal. 25, 342–366 (1967)MathSciNetCrossRefGoogle Scholar
  52. 52.
    T.W. Ting, Elastic-plastic torsion problem III. Arch. Ration. Mech. Anal 34, 228–244 (1969)MathSciNetCrossRefGoogle Scholar
  53. 53.
    R. Von Mises, Three remarks on the theory of the ideal plastic body, in Reissner Anniversary Volume (Edwards, Ann Arbor, 1949)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • G. Colajanni
    • 1
  • Patrizia Daniele
    • 1
    Email author
  • Sofia Giuffrè
    • 2
  • Antonino Maugeri
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of CataniaCataniaItaly
  2. 2.D.I.I.E.S. “Mediterranea” University of Reggio CalabriaReggio CalabriaItaly

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