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An Extension of the Kaliszewski Cone to Non-polyhedral Pointed Cones in Infinite-Dimensional Spaces

  • Lidia Huerga
  • Baasansuren Jadamba
  • Miguel Sama
Article
  • 37 Downloads

Abstract

In this paper, we propose an extension of the family of constructible dilating cones given by Kaliszewski (Quantitative Pareto analysis by cone separation technique, Kluwer Academic Publishers, Boston, 1994) from polyhedral pointed cones in finite-dimensional spaces to a general family of closed, convex, and pointed cones in infinite-dimensional spaces, which in particular covers all separable Banach spaces. We provide an explicit construction of the new family of dilating cones, focusing on sequence spaces and spaces of integrable functions equipped with their natural ordering cones. Finally, using the new dilating cones, we develop a conical regularization scheme for linearly constrained least-squares optimization problems. We present a numerical example to illustrate the efficacy of the proposed framework.

Keywords

Constrained convex optimization Dilating cones Infinite-dimensional analysis Perturbation theory Proper efficiency 

Mathematics Subject Classification

90C20 90C31 90C46 

Notes

Acknowledgements

Baasansuren Jadamba’s work is supported by RITs COS FEAD Grant for 2016-2017 and the National Science Foundation Grant under Award No. 1720067. Lidia Huerga and Miguel Sama’s work is supported by Ministerio de Economía y Competitividad (Spain) under Project MTM2015-68103-P. The authors are very grateful to the anonymous referees for their useful suggestions and remarks.

References

  1. 1.
    Khan, A.A., Sama, M.: A new conical regularization for some optimization and optimal control problems: convergence analysis and finite element discretization. Numer. Funct. Anal. Optim. 34(8), 861–895 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33(2–3), 209–228 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Schiela, A.: Barrier methods for optimal control problems with state constraints. SIAM J. Optim. 20(2), 1002–1031 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D’Apice, C., Kogut, P.I., Manzo, R.: On relaxation of state constrained optimal control problem for a PDE–ODE model of supply chains. Netw. Heterog. Media 9(3), 501–518 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Mas-Colell, A., Zame, W.R.: Equilibrium theory in infinite-dimensional spaces. In: Hildenbrand, W., Sonnenschein, H. (eds.) Handbook of Mathematical Economics, vol. 4, pp. 1835–1898. North-Holland, Amsterdam (1991)Google Scholar
  6. 6.
    Martin, K., Ryan, C.T., Stern, M.: The Slater conundrum: duality and pricing in infinite-dimensional optimization. SIAM J. Optim. 26(1), 111–138 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ghate, A.: Circumventing the Slater conundrum in countably infinite linear programs. Eur. J. Oper. Res. 246(3), 708–720 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36(3), 387–407 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jadamba, B., Khan, A.A., Sama, M.: Regularization for state constrained optimal control problems by half spaces based decoupling. Syst. Control Lett. 61(6), 707–713 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kogut, P.I., Leugering, G., Schiel, R.: On the relaxation of state-constrained linear control problems via Henig dilating cones. Control Cybern. 45(2), 131–162 (2016)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Jadamba, B., Khan, A., Sama, M.: Error estimates for integral constraint regularization of state-constrained elliptic control problems. Comput. Optim. Appl. 67(1), 39–71 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, pp. 481–492. University of California Press, Berkeley and Los Angeles (1951)Google Scholar
  13. 13.
    Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71(1), 232–241 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Borwein, J.: Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15(1), 57–63 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Borwein, J.M., Zhuang, D.: Super efficiency in vector optimization. Trans. Am. Math. Soc. 338(1), 105–122 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gutiérrez, C., Huerga, L., Novo, V.: Nonlinear scalarization in multiobjective optimization with a polyhedral ordering cone. Int. Trans. Oper. Res. 25(3), 763–779 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Daniilidis, A.: Arrow–Barankin–Blackwell theorems and related results in cone duality: a survey. In: Lecture Notes in Economics and Mathematical Systems, vol. 481, pp. 119–131. Springer, Berlin (2000)Google Scholar
  19. 19.
    Sterna-Karwat, A.: Approximating families of cones and proper efficiency in vector optimization. Optimization 20(6), 809–817 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kaliszewski, I.: Quantitative Pareto Analysis by Cone Separation Technique. Kluwer Academic Publishers, Boston (1994)CrossRefzbMATHGoogle Scholar
  21. 21.
    Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 17. Springer, New York (2003)zbMATHGoogle Scholar
  22. 22.
    Henig, M.I.: A cone separation theorem. J. Optim. Theory Appl. 36(3), 451–455 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, 3rd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  24. 24.
    Borwein, J.M., Vanderwerff, J.D.: Constructible convex sets. Set-Valued Anal. 12(1–2), 61–77 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Casini, E., Miglierina, E.: Cones with bounded and unbounded bases and reflexivity. Nonlinear Anal. 72(5), 2356–2366 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Bishop, E., Phelps, R.R.: The support functionals of a convex set. Proc. Symp. Pure Math. 7, 27–35 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Davydov, O.: Approximation by piecewise constants on convex partitions. J. Approx. Theory 164(2), 346–352 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ponstein, J.: On the use of purely finitely additive multipliers in mathematical programming. J. Optim. Theory Appl. 33(1), 37–55 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hohenwarter, M.: GeoGebra: Ein Softwaresystem für dynamische Geometrie und Algebra der Ebene. Master’s thesis, Paris Lodron University, Salzburg, Austria (2002)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Lidia Huerga
    • 1
  • Baasansuren Jadamba
    • 2
  • Miguel Sama
    • 1
  1. 1.Departamento de Matemática AplicadaE.T.S.I. Industriales Universidad Nacional de Educación a DistanciaMadridSpain
  2. 2.Center for Applied and Computational Mathematics, School of Mathematical SciencesRochester Institute of TechnologyRochesterUSA

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