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A Unified Characterization of Nonlinear Scalarizing Functionals in Optimization

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Abstract

Over the years, several classes of scalarization techniques in optimization have been introduced and employed in deriving separation theorems, optimality conditions and algorithms. In this paper, we study the relationships between some of those classes in the sense of inclusion. We focus on three types of scalarizing functionals defined by Hiriart-Urruty, Drummond and Svaiter, and Gerstewitz. We completely determine their relationships. In particular, it is shown that the class of the functionals by Gerstewitz is minimal in this sense. Furthermore, we define a new (and larger) class of scalarizing functionals that are not necessarily convex, but rather quasidifferentiable and positively homogeneous. We show that our results are connected with some of the set relations in set optimization.

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References

  1. Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces MPS/SIAM Series on Optimization, vol. 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)

  2. Aussel, D., Daniilidis, A.: Normal cones to sublevel sets: An axiomatic approach. In: Hadjisavvas, N., Martínez-Legaz, J.E., Penot, J. (eds.) Generalized Convexity and Generalized Monotonicity. Lecture Notes in Economics and Mathematical Systems, vol. 502, pp 88–101. Springer, Berlin (2001)

  3. Briec, W.: Minimum distance to the complement of a convex set: Duality result. J. Optim. Theory Appl. 93, 301–319 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cabot, A., Thibault, L.: Sequential formulae for the normal cone to sublevel sets. Trans. Am. Math. Soc. 366, 6591–6628 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Crespi, G.P., Ginchev, I., Rocca, M.: Points of efficiency in vector optimization with increasing-along-rays property and Minty variational inequalities. In: Konnov, I.V., Luc, D.T., Rubinov, A. (eds.) Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, vol. 583, pp 209–226. Springer, Berlin (2006)

  7. Coulibaly, A., Crouzeix, J.-P.: Condition numbers and error bounds in convex programming. Math. Program. Ser. B 116, 79–113 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  9. Demyanov, V.F., Rubinov, A.M. (eds.): Quasidifferentiability and Related Topics. Nonconvex Optimization and its Applications, vol. 43. Kluwer Academic Publishers, Dordrecht (2000)

  10. Durea, M., Tammer, C.: Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58, 449–467 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dutta, J., Tammer, C: Lagrangian conditions for vector optimization in Banach spaces. Math. Meth. Oper. Res. 64, 521–541 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gao, Y., Yang, X.M.: Properties of the nonlinear scalar functional and its applications to vector optimization problems. J. Glob. Optim. 73, 869–889 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gerstewitz (Tammer), C.: Nichtkonvexe Dualität in der Vektoroptimierung. Wiss. Z. Tech. Hochsch. Leuna-Merseburg 25, 357–364 (1983)

    MathSciNet  MATH  Google Scholar 

  14. Gerstewitz, C., Iwanow, E.: Dualität für nichtkonvexe Vektoroptimierungsprobleme. Wiss. Z. Tech. Hochsch. Ilmenau 31, 61–81 (1985)

    MathSciNet  MATH  Google Scholar 

  15. Gerth, G., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)

    MATH  Google Scholar 

  17. Graña Drummond, L.M., Svaiter, B.F.: A steepest descent method for vector optimization. J. Comput. Appl. Math. 175, 395–414 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gutiérrez, C., Jiménez, B., Miglierina, E., Molho, E.: Scalarization in set optimization with solid and nonsolid ordering cones. J. Glob. Optim. 61, 525–552 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ha, T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. In: Pardalos, P.M., Rassias, T.M., Khan, AA (eds.) Nonlinear Analysis and Variational Problems. Springer Optimization and Its Applications, vol. 35, pp 305–324. Springer, New York (2010)

  20. Hamel, A.H.: A Very Short History of Directional Translative Functions. Draft, Yeshiva University, New York (2012)

    Google Scholar 

  21. Hadwiger, H.: Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt. Math. Z. 53, 210–218 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hiriart-Urruty, J.B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79–97 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hiriart-Urruty, J.-B.: New concepts in nondifferentiable programming. Bull. Soc. Math. France 60, 57–85 (1979)

    MathSciNet  MATH  Google Scholar 

  24. Holmes, R.B.: Geometric Functional Analysis and its Applications. Springer, New York (1975)

    Book  MATH  Google Scholar 

  25. Ioffe, A.D.: Approximate subdifferentials and applications II. Mathematika 33, 111–128 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ioffe, A.D.: Approximate subdifferentials and applications III. The metric theory. Mathematika 36, 1–38 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55, 501–558 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ioffe, A.D.: Variational Analysis of Regular Mappings: Theory and Applications. Springer, Cham (2017)

    Book  MATH  Google Scholar 

  29. Jahn, J.: Bishop–Phelps cones in optimization. Int. J. Optim. Theory Methods Appl. 1, 123–139 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Jahn, J.: Vector Optimization: Theory, Applications, and Extensions, 2nd edn. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  31. Jahn, J.: Vectorization in set optimization. J. Optim. Theory Appl. 167, 783–795 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Jahn, J., Ha, T.X.D.: New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kasimbeyli, R.: A nonlinear cone separation theorem and scalarization in nonconvex vector optimization. SIAM J. Optim. 20, 1591–1619 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kasimbeyli, R., Ozturk, Z.K., Kasimbeyli, N., Yalcin, G.D., Icmen, B.: Conic scalarization method in multiobjective optimization and relations with other scalarization methods. In: Le Thi, H.A., Pham Dinh, T., Nguyen, N. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences. Advances in Intelligent Systems and Computing, vol. 359, pp 319–329. Springer, Cham (2015)

  35. Khan, A., Tammer, C., Zălinescu, C.: Set-valued Optimization: An Introduction with Applications. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  36. Kuroiwa, D.: The natural criteria in set-valued optimization. RIMS Kokyuroku 1031, 85–90 (1980)

    MATH  Google Scholar 

  37. Li, G.H., Li, S.J., You, M.X.: Relationships between the oriented distance functional and a nonlinear separation functional. J. Math. Anal. Appl. 466, 1109–1117 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  38. Miglierina, E., Molho, E.: Scalarization and stability in vector optimization. J. Optim. Theory Appl. 114, 657–670 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Grundlehren der mathematischen Wissenschaften, vol. 330. Springer, Berlin (2006)

  40. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Grundlehren der mathematischen Wissenschaften, vol. 331. Springer, Berlin (2006)

  41. Pontryagin, L.S.: Linear differential games. II. Soviet Math. Dokl. 8, 910–912 (1967)

    MATH  Google Scholar 

  42. Schirotzek, W.: Nonsmooth Analysis. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  43. Wierzbicki, A.P.: The use of reference objectives in multiobjective optimization. In: Fandel, G., Gal, T (eds.) Multiple Criteria Decision Making: Theory and Applications. Lecture Notes in Economics and Mathematical Systems, vol. 177, pp 468–486. Springer, Berlin (1980)

  44. Wierzbicki, A.P.: Basic properties of scalarizing functionals for multiobjective optimization. Optimization 8, 55–60 (1977)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Havana, 10400, Havana, Cuba

    Gemayqzel Bouza

  2. Institute of Mathematics, Martin-Luther-University of Halle-Wittenberg, Theodor-Lieser-Str. 5, 06120, Halle-Saale, Germany

    Ernest Quintana & Christiane Tammer

Authors
  1. Gemayqzel Bouza
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  2. Ernest Quintana
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  3. Christiane Tammer
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Correspondence to Christiane Tammer.

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Dedicated to Alexander Ioffe in honor of his 80th birthday.

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Bouza, G., Quintana, E. & Tammer, C. A Unified Characterization of Nonlinear Scalarizing Functionals in Optimization. Vietnam J. Math. 47, 683–713 (2019). https://doi.org/10.1007/s10013-019-00359-1

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