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

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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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Schiela, A.: Barrier methods for optimal control problems with state constraints. SIAM J. Optim. 20(2), 1002–1031 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ghate, A.: Circumventing the Slater conundrum in countably infinite linear programs. Eur. J. Oper. Res. 246(3), 708–720 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36(3), 387–407 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  13. Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Borwein, J.: Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15(1), 57–63 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  16. Borwein, J.M., Zhuang, D.: Super efficiency in vector optimization. Trans. Am. Math. Soc. 338(1), 105–122 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  19. Sterna-Karwat, A.: Approximating families of cones and proper efficiency in vector optimization. Optimization 20(6), 809–817 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kaliszewski, I.: Quantitative Pareto Analysis by Cone Separation Technique. Kluwer Academic Publishers, Boston (1994)

    Book  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  22. Henig, M.I.: A cone separation theorem. J. Optim. Theory Appl. 36(3), 451–455 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, 3rd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  24. Borwein, J.M., Vanderwerff, J.D.: Constructible convex sets. Set-Valued Anal. 12(1–2), 61–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Casini, E., Miglierina, E.: Cones with bounded and unbounded bases and reflexivity. Nonlinear Anal. 72(5), 2356–2366 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Bishop, E., Phelps, R.R.: The support functionals of a convex set. Proc. Symp. Pure Math. 7, 27–35 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  27. Davydov, O.: Approximation by piecewise constants on convex partitions. J. Approx. Theory 164(2), 346–352 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ponstein, J.: On the use of purely finitely additive multipliers in mathematical programming. J. Optim. Theory Appl. 33(1), 37–55 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hohenwarter, M.: GeoGebra: Ein Softwaresystem für dynamische Geometrie und Algebra der Ebene. Master’s thesis, Paris Lodron University, Salzburg, Austria (2002)

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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.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, E.T.S.I. Industriales Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, 28040, Madrid, Spain

    Lidia Huerga & Miguel Sama

  2. Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY, 14623, USA

    Baasansuren Jadamba

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  1. Lidia Huerga
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  2. Baasansuren Jadamba
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  3. Miguel Sama
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Correspondence to Miguel Sama.

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Communicated by Juan Parra.

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Huerga, L., Jadamba, B. & Sama, M. An Extension of the Kaliszewski Cone to Non-polyhedral Pointed Cones in Infinite-Dimensional Spaces. J Optim Theory Appl 181, 437–455 (2019). https://doi.org/10.1007/s10957-018-01468-6

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