Advertisement

Characterizations of Solution Sets of Differentiable Quasiconvex Programming Problems

  • Vsevolod I. IvanovEmail author
Article

Abstract

In this paper, we study some problems with a continuously differentiable and quasiconvex objective function. We prove that exactly one of the following two alternatives holds: (I) the gradient of the objective function is different from zero over the solution set, and the normalized gradient is constant over it; (II) the gradient of the objective function is equal to zero over the solution set. As a consequence, we obtain characterizations of the solution set of a program with a continuously differentiable and quasiconvex objective function, provided that one of the solutions is known. We also derive Lagrange multiplier characterizations of the solutions set of an inequality constrained problem with continuously differentiable objective function and differentiable constraints, which are all quasiconvex on some convex set, not necessarily open. We compare our results with the previous ones. Several examples are provided.

Keywords

Quasiconvex function Characterizations of the solution set Quasiconvex program Pseudoconvex function KKT Conditions 

Mathematics Subject Classification

90C26 26B25 90C46 

References

  1. 1.
    Mangasarian, O.L.: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7, 21–26 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Burke, J.V., Ferris, M.C.: Characterization of the solution sets of convex programs. Oper. Res. Lett. 10, 57–60 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Jeyakumar, V., Yang, X.Q.: On characterizing the solution sets of pseudolinear programs. J. Optim. Theory Appl. 87, 747–755 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ivanov, V.I.: First-order characterizations of pseudoconvex functions. Serdica Math. J. 27, 203–218 (2001)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ivanov, V.I.: Characterizations of the solution sets of generalized convex minimization problems. Serdica Math. J. 29, 1–10 (2003)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ivanov, V.I.: Optimality conditions and characterizations of the solution sets in generalized convex problems and variational inequalities. J. Optim. Theory Appl. 158, 65–84 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ivanov, V.I.: Characterizations of pseudoconvex functions and semistrictly quasiconvex ones. J. Global Optim. 57, 677–693 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wu, Z.: The convexity of the solution set of a pseudoconvex inequality. Nonlinear Anal. TMA 69, 1666–1674 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Yang, X.M.: On characterizing the solution sets of pseudoinvex extremum problems. J. Optim. Theory Appl. 140, 537–542 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Liu, C., Yang, X., Lee, H.: Characterizations of the solution sets of pseudoinvex programs and variational inequalities. J. Inequal. Appl. 2011, 32 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ivanov, V.I.: Higher order invex functions and higher order pseudoinvex ones. Appl. Anal. 92, 2152–2167 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Smietanski, M.: A note on characterization of solution sets of pseudolinear programming problems. Appl. Anal. 91, 2095–2104 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Barani, A.: Convexity of the solution set of a pseudoconvex inequality in Riemannian manifolds. Numer. Funct. Anal. Optim. 39, 588–599 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jeyakumar, V., Lee, G.M., Dinh, N.: Lagrange multiplier conditions characterizing the optimal solution sets of cone-constrained convex programs. J. Optim. Theory Appl. 123, 83–103 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jeyakumar, V., Lee, G.M., Dinh, N.: Characterizations of solution sets of convex vector minimization problems. Eur. J. Oper. Res. 174, 1380–1395 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dinh, N., Jeyakumar, V., Lee, G.M.: Lagrange multiplier characterizations of solution sets of constrained pseudolinear optimization problems. Optimization 55, 241–250 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Son, T.Q., Dinh, N.: Characterizations of optimal solution sets of convex infinite programs. Top 16, 147–163 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lalitha, C.S., Mehta, M.: Characterizations of the solution sets of mathematical programs in terms of Lagrange multipliers. Optimization 58, 995–1007 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zhao, K.Q., Yang, X.M.: Characterizations of solution set for a class of nonsmooth optimization problems. Optim. Lett. 7, 685–694 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Son, T.Q., Kim, D.S.: A new approach to characterize the solution set of a pseudoconvex programming problem. J. Comput. Appl. Math. 261, 333–340 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Long, X.J., Peng, Z.Y., Wang, X.: Characterizations of the solution set for nonconvex semi-infinite programming problems. J. Nonlinear Convex Anal. 17, 251–265 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Penot, J.-P.: Characterization of solution set of quasiconvex programs. J. Optim. Theory Appl. 117, 627–636 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ivanov, V.I.: Optimization and variational inequalities with pseudoconvex functions. J. Optim. Theory Appl. 146, 602–616 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Suzuki, S., Kuroiwa, D.: Characterizations of solution set for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. J. Global Optim. 62, 431–441 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Suzuki, S., Kuroiwa, D.: Characterizations of the solution set for non-essentially quasiconvex programming. Optim. Lett. 11, 1699–1712 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mangasarian, O.L.: Nonlinear Programming. Classics in Applied Mathematics. SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  27. 27.
    Arrow, K.J., Enthoven, A.C.: Quasi-concave programming. Econometrica 29, 779–800 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gordan, P.: Über die Auflösungen linearer Gleichungen mit reelen coefficienten. Math. Ann. 6, 23–28 (1873)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Bertsekas, D.P., Nedic, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)zbMATHGoogle Scholar
  30. 30.
    Avriel, M., Diewert, W., Schaible, S., Zang, I.: Generalized Concavity, Classics in Applied Mathematics (Originally published: New York (1988)), vol. 63. SIAM, Philadelphia (2010)Google Scholar
  31. 31.
    Greenberg, H.J., Pierskalla, W.P.: Quasiconjugate functions and surrogate duality. Cah. Cent. Etud. Rech. Oper. 15, 437–448 (1973)zbMATHGoogle Scholar
  32. 32.
    Martinez-Legaz, J.E.: Quasiconvex duality theory by generalized conjugation methods. Optimization 19, 603–652 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Technical University of VarnaVarnaBulgaria

Personalised recommendations