Optimization Letters

, Volume 7, Issue 5, pp 903–911 | Cite as

Necessary and sufficient condition on global optimality without convexity and second order differentiability

  • Pál BuraiEmail author
Original Paper


The main goal of this paper is to give a necessary and sufficient condition of global optimality for unconstrained optimization problems, when the objective function is not necessarily convex. We use Gâteaux differentiability of the objective function and its bidual (the latter is known from convex analysis).


Banach Space Stationary Point Nonempty Subset Convex Analysis Unconstrained Optimization Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bonnans J.F., Shapiro A.: Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer, New York (2000)Google Scholar
  2. 2.
    Brøndsted A., Rockafellar R.T.: On the subdifferentiability of convex functions. Proc. Am. Math. Soc. 16, 605–611 (1965)Google Scholar
  3. 3.
    Clarke F.H.: Optimization and nonsmooth analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1983)Google Scholar
  4. 4.
    Ekeland I.: Nonconvex minimization problems. Bull. Am. Math. Soc. (N.S.) 1(3), 443–474 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Ekeland, I., Temam, R.: Convex analysis and variational problems. Studies in Mathematics and its Applications, vol. 1. North-Holland Publishing Co./American Elsevier Publishing Co., Inc., Amsterdam/New York (1976) (translated from the French)Google Scholar
  6. 6.
    Georgiev P.G., Chinchuluun A., Pardalos P.M.: Optimality conditions of first order for global minima of locally Lipschitz functions. Optimization 60(1–2), 277–282 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Hiriart-Urruty J.B.: When is a point x satisfying ∇ f(x) = 0 a global minimum of f?. Am. Math. Mon. 93(7), 556–558 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ioffe A.D.: Necessary and sufficient conditions for a local minimum. III. Second order conditions and augmented duality. SIAM. J. Control Optim. 17(2), 266–288 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ioffe A.D., Tihomirov V.M.: Theory of extremal problems. Nort-Holland, Amsterdam (1979)zbMATHGoogle Scholar
  10. 10.
    Rockafellar R.T.: Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton (1970)Google Scholar
  11. 11.
    Tröltzsch, F.: Optimal control of partial differential equations. Theory, methods and applications. Graduate Studies in Mathematics, 112. American Mathematical Society, Providence (2010) (translated from the 2005 German original by Jürgen Sprekels)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Faculty of InformaticsUniversity of DebrecenDebrecenHungary
  2. 2.Department of MathematicsTU BerlinBerlinGermany

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