Set-Valued Analysis

, Volume 12, Issue 1–2, pp 49–60 | Cite as

A One Perturbation Variational Principle and Applications

  • Jonathan Borwein
  • Lixin Cheng
  • Marián Fabian
  • Julian P. Revalski


We study a variational principle in which there is one common perturbation function ϕ for every proper lower semicontinuous extended real-valued function f defined on a metric space X. Necessary and sufficient conditions are given in order for the perturbed function f+ϕ to attain its minimum. In the case of a separable Banach space we obtain a specific principle in which the common perturbation function is, in addition, also convex and Hadamard-like differentiable. This allows us to provide applications of the principle to differentiability of convex functions on separable and more general Banach spaces.

variational principle well-posed optimization problem perturbed optimization problem separable Banach space weak Asplund space Gâteaux differentiability space 


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Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Jonathan Borwein
    • 1
  • Lixin Cheng
    • 2
  • Marián Fabian
    • 3
  • Julian P. Revalski
    • 4
  1. 1.CECM, Department of MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.Department of MathematicsXiamen UniversityXiamenChina
  3. 3.Mathematical InstituteCzech Academy of SciencesPragueCzech Republic
  4. 4.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

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