Exact Penalty in Constrained Optimization and the Mordukhovich Basic Subdifferential

Part of the Springer Optimization and Its Applications book series (SOIA, volume 47)


In this chapter, we use the penalty approach to study two constrained minimization problems in infinite-dimensional Asplund spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. We use the notion of the Mordukhovich basic subdifferential and show that the exact penalty property is stable under perturbations of objective functions.


Penalty Function SIAM Journal Exact Penalty Exact Penalty Function Constrain Minimization Problem 
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  1. 1.
    Bao, T. Q. and Mordukhovich, B. S.: Variational principles for set-valued mappings with applications to multiobjective optimization, Control and Cybernetics 36, 531–562 (2007).MATHMathSciNetGoogle Scholar
  2. 2.
    Bao, T. Q. and Mordukhovich, B. S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions, Mathematical Programming 122(2), 301–347 (2010).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boukari, D. and Fiacco, A. V.: Survey of penalty, exact-penalty and multiplier methods from 1968 to 1993, Optimization 32, 301–334 (1995).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Burke, J. V.: Calmness and exact penalization, SIAM Journal on Control and Optimization 29, 493–497 (1991).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Burke, J. V.: An exact penalization viewpoint of constrained optimization, SIAM Journal on Control and Optimization 29, 968–998 (1991).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Clarke, F. H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983).MATHGoogle Scholar
  7. 7.
    Di Pillo, G. and Grippo, L.: Exact penalty functions in constrained optimization, SIAM Journal on Control and Optimization 27, 1333–1360 (1989).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ekeland, I.: On the variational principle, Journal of Mathematical Analysis and Applications 47, 324–353 (1974).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Eremin, I. I.: The penalty method in convex programming, Soviet Mathematics Doklady 8, 459–462 (1966).Google Scholar
  10. 10.
    Kelley, J. L.: General Topology, D. Van Nostrand, New York (1955).MATHGoogle Scholar
  11. 11.
    Khaladi, M. and Penot, J.-P.: Estimates of the exact penalty coefficient threshold, Utilitas Mathematica 42, 147–161 (1992).MATHMathSciNetGoogle Scholar
  12. 12.
    Mordukhovich, B. S.: Maximum principle in problems of time optimal control with nonsmooth constraints, Journal of Applied Mathematics and Mechanics 40, 960–969 (1976).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006).Google Scholar
  14. 14.
    Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006).Google Scholar
  15. 15.
    Palais, R: Lusternik–Schnirelman theory of Banach manifolds, Topology 5, 115–132 (1966).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sard, A.: The measure of the critical values of differentiable maps, Bulletin of the American Mathematical Society 48, 883–890 (1942).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ward, D. E.: Exact penalties and sufficient conditions for optimality in nonsmooth optimization, Journal Optimization Theory & Applications 57, 485–499 (1988).MATHCrossRefGoogle Scholar
  18. 18.
    Zangwill, W. I.: Nonlinear programming via penalty functions, Management Science 13, 344–358 (1967).CrossRefMathSciNetGoogle Scholar
  19. 19.
    Zaslavski, A. J.: On critical points of Lipschitz functions on smooth manifolds, Siberian Mathematical Journal 22, 63–68 (1981).CrossRefGoogle Scholar
  20. 20.
    Zaslavski, A. J.: A sufficient condition for exact penalty in constrained optimization, SIAM Journal on Optimization 16, 250–262 (2005).MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Zaslavski, A. J.: Exact penalty in constrained optimization and critical points of Lipschitz functions, Journal of Nonlinear and Convex Analysis 10, 149–156 (2009).MATHMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsThe Technion-Israel Institute of TechnologyHaifaIsrael

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