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Vector Variational Principles for Set-Valued Functions

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Recent Developments in Vector Optimization

Part of the book series: Vector Optimization ((VECTOROPT,volume 1))

Abstract

Deriving existence results and necessary conditions for approximate solutions of nonlinear optimization problems under week assumptions is an interesting and modern field in optimization theory. It is of interest to show corresponding results for optimization problems without any convexity and compactness assumptions. Ekeland’s variational principle is a very deep assertion about the existence of an exact solution of a slightly perturbed optimization problem in a neighborhood of an approximate solution of the original problem. The importance of Ekeland’s variational principle in nonlinear analysis is well known. Especially, this assertion is very useful for deriving necessary conditions under certain differentiability assumptions. In optimal control Ekeland’s principle can be used in order to prove an ε-maximum principle in the sense of Pontryagin and in approximation theory for deriving ε-Kolmogorov conditions.

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References

  1. Al-Homidan, S., Ansari, Q.H., Yao, J.-C.: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory, Nonlinear Analysis, Theory, Methods and Applications, 69(1), 126–139 (2008).

    Google Scholar 

  2. Ansari, Q.H.: Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory, Journal of Mathematical Analysis and Applications, 334(2), 561–575 (2007).

    Article  Google Scholar 

  3. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)

    Google Scholar 

  4. Bonnisseau, J.-M., Crettez, B.: On the characterization of efficient production vectors. Economic Theory 31, 213–223 (2007)

    Article  Google Scholar 

  5. Attouch, H., Riahi, H.: Stability results for Ekeland’s ε-variational principle and cone extremal solutions. Math. Oper. Res. 18, 173–201 (1993)

    Article  Google Scholar 

  6. Bao, T.Q., Mordukhovich, B.S.: Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cybernet. 36, 531–562 (2007)

    Google Scholar 

  7. Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Prog. 122, 301–347 (2010)

    Article  Google Scholar 

  8. Bednarczuk, E.M., Zagrodny, D.: Vector variational principle. Arch. Math. (Basel) 93, 577–586 (2009)

    Google Scholar 

  9. Bianchi, M., Kassay, G., Pini, R.: Ekeland’s principle for vector equilibrium problems. Nonlinear Anal. 66, 1454–1464 (2007)

    Article  Google Scholar 

  10. Bonnisseau, J.-M., Cornet, B.: Existence of equilibria when firms follow bounded losses pricing rules. J. Math. Econom. 17, 119–147 (1988)

    Article  Google Scholar 

  11. Borwein, J.M., Preiss, D.: A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. Amer. Math. Soc. 303, 517–527 (1987)

    Article  Google Scholar 

  12. Brøndsted, A.: On a lemma of Bishop and Phelps. Pacific J. Math. 55, 335–341 (1974)

    Google Scholar 

  13. Chen, G.Y., Huang, X.X.: A unified approach to the existing three types of variational principles for vector valued functions. Math. Meth. Oper. Res. 48, 349–357 (1998)

    Article  Google Scholar 

  14. Chen, G.Y., Huang, X.X.: Ekeland’s ε-variational principle for set-valued mappings. Math. Meth. Oper. Res. 48, 181–186 (1998)

    Article  Google Scholar 

  15. Chen, G.Y., Huang, X.X., Lee, G.M.: Equivalents of an approximate variational principle for vector-valued functions and applications. Math. Meth. Oper. Res. 49, 125–136 (1999)

    Google Scholar 

  16. J. Danes̆: A geometric theorem useful in nonlinear functional analysis. Boll. Un. Mat. Ital. 6, 369–375 (1972)

    Google Scholar 

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

    Google Scholar 

  18. Deville, R., El Haddad, E.: The viscosity subdifferential of the sum of two functions in Banach spaces I: First order case. J. Convex Anal. 3, 259–308 (1996)

    Google Scholar 

  19. Dolecki, S., Malivert, C.: General duality in vector optimization. Optimization 27, 97–119 (1993)

    Google Scholar 

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

    Google Scholar 

  21. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  Google Scholar 

  22. Ekeland, I.: Nonconvex minimization problems. Bull. Amer. Math. Soc. 1, 443–474 (1979)

    Article  Google Scholar 

  23. Fabian, M.: Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss. Acta. Univ. Carolin. 30, 51–56 (1989)

    Google Scholar 

  24. Ferro, F.: A new ABB theorem in Banach spaces. Optimization 46, 353–362 (1999)

    Article  Google Scholar 

  25. Figueiredo, D.G. de: Lectures on the Ekeland’s Variational Principle with Applications and Detours. Tata Res. Inst. Bombay. Springer, Berlin (1989)

    Google Scholar 

  26. Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance and Stochastics 4, 429–447 (2002)

    Article  Google Scholar 

  27. Georgiev, P.G.: The strong Ekeland variational principle, the strong drop theorem and applications. J. Math. Anal. Appl. 131, 1–21 (1988)

    Article  Google Scholar 

  28. Gerstewitz (Tammer), Chr., Iwanow, E.: Dualität für nichtkonvexe Vektoroptimierungsprobleme. Wissenschaftliche Zeitschrift der Tech. Hochschule Ilmenau. 31(2), 61–81 (1985)

    Google Scholar 

  29. Göpfert, A., Tammer, Chr.: ε-approximate solutions and conical support points. A new maximal point theorem. Z.A.M.M. 75, 595–596 (1995)

    Google Scholar 

  30. Göpfert, A., Tammer, Chr.: A new maximal point theorem. Z. Anal. Anwendungen 14, 379–390 (1995)

    Google Scholar 

  31. Göpfert, A., Tammer, Chr.: Maximal point theorems in product spaces and applications for multicriteria approximation problems. In: Research and Practice in Multiple Criteria Decision Making. Haimes, Y.Y. and Steuer R.E. (eds.). Lecture Notes in Econom. and Math. Systems. Springer, Berlin 487pp. 93–104 (1999)

    Google Scholar 

  32. Göpfert, A., Riahi, H., Tammer, Chr., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books in Mathematics 17, Springer, New York (2003)

    Google Scholar 

  33. Göpfert, A., Tammer, Chr., Zălinescu, C.: A new minimal point theorem in product spaces. Z. Anal. Anwendungen 18, 767–770 (1999)

    Google Scholar 

  34. Göpfert, A., Tammer, Chr., Zălinescu, C.: On the vectorial Ekeland’s variational principle and minimal points in product spaces. Nonlinear Anal. 39, 909–922 (2000)

    Google Scholar 

  35. Gorokhovik, V.V., Gorokhovik, S. Ya.: A criterion of the global epi-Lipschitz property of sets (Russian). Izvestiya Natsional’no Akademii Nauk Belarusi. Seriya Fiziko-Matematicheskikh Nauk, 118–120 (1995)

    Google Scholar 

  36. Gutiérrez, C., Jiménez, B., Novo, V.: A set-valued Ekeland’s variational principle in vector optimization. SIAM J. Control Optim. 47, 883–903 (2008)

    Article  Google Scholar 

  37. Ha, T.X.D.: Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl. 124, 187–206 (2005)

    Article  Google Scholar 

  38. Hamel, A.: Phelps’ lemma, Danes’ drop theorem and Ekeland’s principle in locally convex topological vector spaces. Proc. Amer. Math. Soc. 131, 3025–3038 (2003)

    Google Scholar 

  39. Hamel, A.: From real to set-valued coherent risk measures. Reports of the Institute of Optimization and Stochastics, Martin-Luther-University Halle-Wittenberg. No. 19, 10–22 (2004)

    Google Scholar 

  40. Hamel, A.: Translative sets and functions and its applications to risk measure theory and nonlinear separation. Preprint, Halle (2007)

    Google Scholar 

  41. Hamel, A., Tammer, Chr.: Minimal elements for product orders, Optimization 57, 263–275 (2008)

    Google Scholar 

  42. Heyde, F.: Coherent risk measures and vector optimization. In: Multicriteria Decision Making and Fuzzy Systems. Küfer, K.-H., Rommelfanger, H., Tammer, Chr. and Winkler, K. (eds). SHAKER Verlag, pp. 3–12 (2006)

    Google Scholar 

  43. Ioffe, A.D.: Metric regularity and subdifferential calculus. Russian Math. Survey 55, 501–558 (2000)

    Article  Google Scholar 

  44. Ioffe, A.D., Penot, J.-P.: Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings. Serdica Math. J. 22, 359–384 (1996)

    Google Scholar 

  45. Isac, G.: The Ekeland principle and the Pareto-efficiency. In: Multiobjective Programming and Goal Programming, Theory and Applications. Tamiz, M. (ed.). Lecture Notes in Economics and Mathematical Systems. Vol. 432, pp. 148–163. Springer, Berlin (1996)

    Google Scholar 

  46. Isac, G., Tammer, Chr.: Nuclear and full nuclear cones in product spaces: Pareto efficiency and an Ekeland type variational principle. Positivity 9, 511–539 (2005)

    Google Scholar 

  47. Iwanow, E., Nehse, R.: On proper efficient solutions of multi-criterial problems (Russian). Wissenschaftliche Zeitschrift Tech. Hochschule Ilmenau 30(5), 55–60 (1984)

    Google Scholar 

  48. Jules, F.: Sur la somme de sous-différentiels de fonctions semi-continues inférieurement. Dissertationes Math. 423(2003)

    Google Scholar 

  49. Li, S.J., Yang, X.Q., Chen, G.Y.: Vector Ekeland variational principle. In: Vector varational inequalities and vector equilibria. Nonconvex optimization and its applications. Giannessi F. (ed.) Vol. 38, pp. 321–333. Kluwer, Dordrecht, (2000)

    Google Scholar 

  50. Luenberger, D.G.: New optimality principles for economic efficiency and equilibrium. J. Optim. Theory Appl. 75, 221–264 (1992)

    Article  Google Scholar 

  51. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications. Springer, Berlin (2006)

    Google Scholar 

  52. Penot, J.-P.: The drop theorem, the petal theorem and Ekeland’s variational principle. Nonlinear Anal. 10, 813–822 (1986)

    Article  Google Scholar 

  53. Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes Math. 1364Springer, Berlin (1989)

    Google Scholar 

  54. Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability (2nd ed.). Lecture Notes Math. 1364. Springer, Berlin (1993)

    Google Scholar 

  55. Rockafellar, R.T.: Clarke’s tangent cones and the boundaries of closed sets in ℝ n. Nonlinear Anal.: Theory, Meth. Appl. 3, 145–154 (1979)

    Google Scholar 

  56. Rockafellar, R.T.: The Theory of Subgradients and its Applications to Problems of Optimization. Convex and Nonconvex Functions. Heldermann, Berlin (1981)

    Google Scholar 

  57. Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Deviation measures in risk analysis and optimization. Finance Stochastics 10, 51–74 (2006)

    Article  Google Scholar 

  58. Rolewicz, S.: On drop property. Studia Math. 85, 27–35 (1987)

    Google Scholar 

  59. Rubinov, A.M., Singer, I.: Topical and sub-topical functions. Downward sets and abstract convexity. Optimization 50, 307–351 (2001)

    Google Scholar 

  60. Ruszczynski, A., Shapiro, A.: Optimization of convex risk measures. Math. Oper. Res. 31, 433–452 (2006)

    Article  Google Scholar 

  61. Tammer, Chr.: A generalization of Ekeland’s variational principle. Optimization 25, 129–141 (1992)

    Google Scholar 

  62. Tammer, Chr.: A variational principle and a fixed point problem. In: Henry, J., Yvon, J.-P. (eds) System Modelling and Optimization. Lecture Notes in Control and Inform. Sci., 197, Springer, London, pp. 248–257 (1994)

    Google Scholar 

  63. Tammer, Chr., Zălinescu, C.: Lipschitz properties of the scalarization function and applications. Optimization 59(2), 305-319 (2010)

    Google Scholar 

  64. Tammer, Chr., Zălinescu, C.: Vector variational principles for set-valued functions. Report Institut fr Mathematik, Report No. 17, 2009. Accepted: Optimization.

    Google Scholar 

  65. Turinici, M.: Maximal elements in a class of order complete metric spaces. Math. Japonica 25, 511–517 (1980)

    Google Scholar 

  66. Zălinescu, C.: On two notions of proper efficiency. In: Optimization in mathematical physics, Pap. 11th Conf. Methods Techniques Math. Phys., Oberwolfach/Ger. 1985, Methoden Verfahren Math. Phys., 34, 77–86 (1987)

    Google Scholar 

  67. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, University Halle–Wittenberg, 06099, Halle, Germany

    Christiane Tammer

  2. Faculty of Mathematics, University Al.I.Cuza Iaşi, 700506, Iaşi, Romania

    Constantin Zălinescu

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  1. Christiane Tammer
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  2. Constantin Zălinescu
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Correspondence to Christiane Tammer .

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  1. Aligarh, 202001, Uttar Pradesh, India

    Qamrul Hasan Ansari

  2. , Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, 80424, Taiwan R.O.C.

    Jen-Chih Yao

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Tammer, C., Zălinescu, C. (2012). Vector Variational Principles for Set-Valued Functions. In: Ansari, Q., Yao, JC. (eds) Recent Developments in Vector Optimization. Vector Optimization, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21114-0_11

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