Taiwanese Journal of Mathematics

STABILITY OF EXACT PENALTY FOR CLASSES OF CONSTRAINED MINIMIZATION PROBLEMS IN BANACH SPACES

Alexander J. Zaslavski

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Abstract

In this paper we use the penalty approach in order to study two constrained minimization problems in Banach spaces. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property holds and is stable under perturbations of objective functions, constraint functions and the right-hand side of constraints.

Article information

Source
Taiwanese J. Math., Volume 12, Number 6 (2008), 1493-1510.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405036

Digital Object Identifier
doi:10.11650/twjm/1500405036

Mathematical Reviews number (MathSciNet)
MR2444868

Zentralblatt MATH identifier
1155.49018

Subjects
Primary: 49M30: Other methods 90C26: Nonconvex programming, global optimization 90C30: Nonlinear programming

Keywords
Clarke's generalized gradient Ekeland's variational principle minimization problem penalty function

Citation

Zaslavski, Alexander J. STABILITY OF EXACT PENALTY FOR CLASSES OF CONSTRAINED MINIMIZATION PROBLEMS IN BANACH SPACES. Taiwanese J. Math. 12 (2008), no. 6, 1493--1510. doi:10.11650/twjm/1500405036. https://projecteuclid.org/euclid.twjm/1500405036


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References

  • [1.] T. Q. Bao and B. S. Mordukhovich, Variational principles for set-valued mappings with applications to multiobjective optimization, Control and Cybernetics, 36 (2007).
  • [2.] D. Boukari and A. V. Fiacco, Survey of penalty, exact-penalty and multiplier methods from 1968 to 1993 Optimization, 32 (1995), 301-334.
  • [3.] J. V. Burke, An exact penalization viewpoint of constrained optimization, SIAM J. Control Optim., 29 (1991), 968-998.
  • [4.] F. H. Clarke, Optimization and Nonsmooth Analysis, Willey Interscience, 1983.
  • [5.] G. Di Pillo and L. Grippo, Exact penalty functions in constrained optimization, SIAM J. Control Optim., 27 (1989), 1333-1360.
  • [6.] I. Ekeland, On the variational principle J. Math. Anal. Appl., 47 (1974), 324-353.
  • [7.] I. I. Eremin, The penalty method in convex programming, Soviet Math. Dokl., 8 (1966), 459-462.
  • [8.] S.-P. Han and O. L. Mangasarian, Exact penalty function in nonlinear programming, Math. Programming, 17 (1979), 251-269.
  • [9.] J.-B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993.
  • [10.] D. G. Luenberger, Control problems with kinks IEEE Trans. Automat. Control, 15 (1970), 570-575.
  • [11.] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications, Springer, Berlin, 2006.
  • [12.] J. Outrata, A note on the usage of nondifferentiable exact penalties in some special optimization problems, Kybernetika, 24 (1988), 251-258.
  • [13.] R. T. Rockafellar, Penalty methods and augmented Lagrangians in nonlinear programming, Fifth Conference on Optimization Techniques (Rome, 1973), Part I. Lecture Notes in Comput. Sci., Springer, Berlin, 3, 1973, pp. 418-425.
  • [14.] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998.
  • [15.] W. I. Zangwill, Nonlinear programming via penalty functions, Management Sci., 13 (1967), 344-358.
  • [16.] A. J. Zaslavski, On critical points of Lipschitz functions on smooth manifolds, Siberian Math. J., 22 (1981), 63-68.
  • [17.] A. J. Zaslavski, A sufficient condition for exact penalty in constrained optimization, SIAM Journal on Optimization, 16 (2005), 250-262.
  • [18.] A. J. Zaslavski, Existence of exact penalty and its stability for nonconvex constrained optimization problems in Banach spaces, Set-Valued Analysis, in press.
  • [19.] A. J. Zaslavski, Stability of exact penalty for nonconvex inequality-constrained minimization problems, Taiwanese J. Math., in accepted.