Set-Valued Systems with Infinite-Dimensional Image and Applications

  • J. Li
  • L. Yang


In infinite-dimensional spaces, we investigate a set-valued system from the image perspective. By exploiting the quasi-interior and the quasi-relative interior, we give some new equivalent characterizations of (proper, regular) linear separation and establish some new sufficient and necessary conditions for the impossibility of the system under new assumptions, which do not require the set to have nonempty interior. We also present under mild assumptions the equivalence between (proper, regular) linear separation and saddle points of Lagrangian functions for the system. These results are applied to obtain some new saddle point sufficient and necessary optimality conditions of vector optimization problems.


Image space analysis Generalized system Set-valued mapping Quasi-relative interior Saddle point 

Mathematics Subject Classification

90C29 90C46 



The authors greatly appreciate three anonymous referees for their useful comments and suggestions, which have helped to improve an early version of the paper. The authors also greatly appreciate Dr. G. Mastroeni for his valuable suggestions on the last part of Sect. 5. The research was supported by the National Natural Science Foundation of China, Project 11371015; the Key Project of Chinese Ministry of Education, Project 211163; and Sichuan Youth Science and Technology Foundation, Project 2012JQ0032.


  1. 1.
    Giannessi, F.: Constrained Optimization and Image Space Analysis. Springer, New York (2005)MATHGoogle Scholar
  2. 2.
    Giannessi, F., Rapcsák, T.: Images, separation of sets and extremum problems. In: Recent Trends in Optimization Theory and Applications. World Scientific Series in Applicable Analysis vol. 5, pp. 79–106 (1995)Google Scholar
  3. 3.
    Li, J., Mastroeni, G.: Image convexity of generalized systems with infinite dimensional image and applications. J. Optim. Theory Appl. 169, 91–115 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Mastroeni, G., Rapcsák, T.: On convex generalized systems. J. Optim. Theory Appl. 104, 605–627 (2000)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Castellani, M., Mastroeni, G., Pappalardo, M.: On regularity for generalized systems and applications. In: Di Pillo, G., Giannessi, F. (eds.) Nonlinear Optimization and Applications, pp. 13–26. Plenum Publishing Corporation, New York (1996)CrossRefGoogle Scholar
  6. 6.
    Guu, S.M., Li, J.: Vector quasi-equilibrium problems: separation, saddle points and error bounds for the solution set. J. Global Optim. 58, 751–767 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Li, J., Feng, S.Q., Zhang, Z.: A unified approach for constrained extremum problems: image space analysis. J. Optim. Theory Appl. 159, 69–92 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Li, J., Huang, N.J.: Image space analysis for variational inequalities with cone constraints and applications to traffic equilibria. Sci. China Math. 55, 851–868 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems. Part I: image space analysis. J. Optim. Theory Appl. 161, 738–762 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems. Part II: special duality schemes. J. Optim. Theory Appl. 161, 763–782 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Carathéodory, M.: Calculus of Variations and Partial Differential Equations of the First Order. Chelsea Publ. Co., New York (1982). Translation of the volume “Variationsrechnung und Partielle Differential Gleichungen Erster Ordnung”. B.G. Teubner, Berlin (1935)Google Scholar
  12. 12.
    Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)MATHGoogle Scholar
  13. 13.
    Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331–365 (1984)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Castellani, G., Giannessi, F.: Decomposition of mathematical programs by means of theorems of alternative for linear and nonlinear systems. In: Survey of Mathematical Programming (Proceedings of the Ninth International Mathematical Programming Symposium. Budapest, 1976), vol. 2, pp. 423–439. North-Holland, Amsterdam (1979)Google Scholar
  15. 15.
    Hestenes, M.R.: Optimization Theory: The Finite Dimensional Case. Wiley, New York (1975)MATHGoogle Scholar
  16. 16.
    Borwein, J.M., Lewis, A.S.: Partially finite convex programming, part I: quasi-relative interiors and duality theory. Math. Program. Ser. B. 57, 15–48 (1992)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Daniele, P., Giuffrè, S., Idone, G., Maugeri, A.: Infinite dimensional duality and applications. Math. Ann. 339, 221–239 (2007)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Maugeri, A., Raciti, F.: Remarks on infinite dimensional duality. J. Global Optim. 46, 581–588 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Limber, M.A., Goodrich, R.K.: Quasi onteriors, Lagrange multipliers, and \(L^p\) spectral estimation with lattice bounds. J. Optim. Theory Appl. 78, 143–161 (1993)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Boţ, R.I., Csetnek, E.R., Wanka, G.: Regularity conditions via quasi-relative interior in convex programming. SIAM J. Optim. 19, 217–233 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Boţ, R.I., Csetnek, E.R., Moldovan, A.: Revisiting some duality theorems via the quasirelative interior in convex optimization. J. Optim. Theory Appl. 139, 67–84 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Cammaroto, F., Di Bella, B.: Separation theorem based on the quasirelative interior and application to duality theory. J. Optim. Theory Appl. 125, 223–229 (2005)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Flores-Bazán, F., Mastroeni, G.: Strong duality in cone constrained nonconvex optimization. SIAM J. Optim. 23, 153–169 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Tasset, T. N.: Lagrange multipliers for set-valued functions when ordering cones have empty interior. Thesis (Ph.D.), University of Colorado at Boulder. ProQuest LLC, Ann Arbor, MI (2010)Google Scholar
  25. 25.
    Zălinescu, C.: On the use of the quasi-relative interior in optimization. Optimzation 64, 1795–1823 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Borwein, J.M., Goebel, R.: Notions of relative interior in Banach spaces. J. Math. Sci. (N. Y.) 115, 2542–2553 (2003)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefMATHGoogle Scholar
  28. 28.
    Jahn, J.: Vector Optimization. Theory, Applications, and Extensions. Springer, Berlin (2011)MATHGoogle Scholar
  29. 29.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002)CrossRefMATHGoogle Scholar
  30. 30.
    Giannessi, F.: Theorems of the alternative for multifunctions with applications to optimization: general results. J. Optim. Theory Appl. 55, 233–256 (1987)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Borwein, J.M.: A multivalued approach to the Farkas lemma. Point-to-set maps and mathematical programming. Math. Programm. Stud. 10, 42–47 (1979)CrossRefMATHGoogle Scholar
  32. 32.
    Borwein, J.M.: Multivalued convexity and optimization: a unified approach to inequality and equality constraints. Math. Programm. 13, 183–199 (1977)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization. An introduction with Applications. Vector Optimization. Springer, Heidelberg (2015)MATHGoogle Scholar
  34. 34.
    Zhou, Z.A., Yang, X.M.: Optimality conditions of generalized subconvex like set-valued optimization problems based on the quasi-relative interior. J. Optim. Theory Appl. 150, 327–340 (2011)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Guu, S.M., Huang, N.J., Li, J.: Scalarization approaches for set-valued vector optimization problems and vector variational inequalities. J. Math. Anal. Appl. 356, 564–576 (2009)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Schaefer, H.H.: Topological Vector Spaces. Springer, New York (1980)MATHGoogle Scholar
  37. 37.
    Breckner, W.W., Kassay, G.: A systematization of convexity concepts for sets and functions. J. Convex Anal. 4, 109–127 (1997)MathSciNetMATHGoogle Scholar
  38. 38.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefMATHGoogle Scholar
  39. 39.
    Flores-Bazán, F., Flores-Bazán, F., Vera, C.: A complete characterization of strong duality in nonconvex optimization with a single constraint. J. Global Optim. 53, 185–201 (2012)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Giannessi, F., Mastroeni, G.: Separation of sets and Wolfe duality. J. Global Optim. 42, 401–412 (2008)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Holmes, R.B.: Geometric Functional Analysis and Its Applications. Springer, New York (1975)CrossRefMATHGoogle Scholar
  42. 42.
    Jabri, Y.: The Mountain Pass Theorem. Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar
  43. 43.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  44. 44.
    Giannessi, F., Mastroeni, G., Yao, J.C.: On maximum and variational principles via image space analysis. Positivity 16, 405–427 (2012)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Courant, R.: Dirichlet Principle, Conformal Mappings and Minimal Surfaces. Interscience, New York (1950)MATHGoogle Scholar
  46. 46.
    Shi, S.Z.: Convex Analysis. Shanghai Science and Technology Press, Shanghai (1990)Google Scholar
  47. 47.
    You, Z.Y., Gong, H.Y., Xu, Z.B.: Nonlinear Analysis. Xi’an Jiaotong University Press, Xi’an (1991)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.College of Mathematics and InformationChina West Normal UniversityNanchongChina

Personalised recommendations