Journal of Optimization Theory and Applications

, Volume 64, Issue 1, pp 141–152 | Cite as

Image space approach to penalty methods

  • M. Pappalardo
Contributed Papers


In this paper, we introduce a unified framework for the study of penalty concepts by means of the separation functions in the image space (see Ref. 1). Moreover, we establish new results concerning a correspondence between the solutions of the constrained problem and the limit points of the unconstrained minima. Finally, we analyze some known classes of penalty functions and some known classical results about penalization, and we show that they can be derived from our results directly.

Key Words

Penalty functions image space separation functions nonlinear programming constrained optimization 


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  1. 1.
    Giannessi, F.,Theorems of Alternative and Optimality Conditions, Journal of Optimization Theory and Applications, Vol. 42, pp. 331–365, 1984.Google Scholar
  2. 2.
    Giannessi, F.,Metodi Matematici della Programmazione. Problemi Lineari e Non Lineari, Quaderni dell'Unione Matematica Italiana, Pitagora Editrice, Bologna, Italy, 1982.Google Scholar
  3. 3.
    Auslender, A.,Penalty Methods for Computing Points That Satisfy Second-Order Necessary Conditions, Mathematical Programming, Vol. 17, pp. 229–238, 1979.Google Scholar
  4. 4.
    Bertsekas, D. P.,Necessary and Sufficient Conditions for a Penalty Method to Be Exact, Mathematical Programming, Vol. 9, pp. 87–99, 1975.Google Scholar
  5. 5.
    Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, New York, 1968.Google Scholar
  6. 6.
    Minoux, M.,Mathematical Programming, Wiley, New York, New York, 1987.Google Scholar
  7. 7.
    Bartholomew-Biggs, M. C.,Recursive Quadratic Programming Methods Based on the Augmented Lagrangian, Mathematical Programming Study, Vol. 31, pp. 21–41, 1987.Google Scholar
  8. 8.
    Bertsekas, D. P.,Constrained Optimization and Lagrange Multipliers Methods, Academic Press, New York, New York, 1982.Google Scholar
  9. 9.
    Szego, G. P.,Programmazione Matematica, Minimization Algorithms, Edited by G. P. Szego, Academic Press, New York, New York, 1972.Google Scholar
  10. 10.
    Fletcher, R.,Penalty Functions, Mathematical Programming Symposium: The State of the Art, Bonn, Germany, 1982; Springer-Verlag, Berlin, Germany, 1982.Google Scholar
  11. 11.
    Di Pillo, G., andGrippo, L.,A Continuously Differentiable Exact Penalty Function for Nonlinear Programming Problems with Inequality Constraints, SIAM Journal on Control and Optimization, Vol. 23, pp. 72–84, 1985.Google Scholar
  12. 12.
    Han, S. P., andMangasarian, O. L.,Exact Penalty Functions in Nonlinear Programming, Mathematical Programming, Vol. 17, pp. 251–265, 1979.Google Scholar
  13. 13.
    Powell, M. J.,Algorithms for Nonlinear Constraints That Use Lagrangian Functions, Mathematical Programming, Vol. 14, pp. 224–248, 1978.Google Scholar
  14. 14.
    Rockafellar, R. T.,Penalty Methods and Augmented Lagrangians in Nonlinear Programming, Proceedings of the 5th IFIP Conference on Optimization Techniques, Rome, Italy, 1973.Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • M. Pappalardo
    • 1
  1. 1.Department of MathematicsUniversity of PisaPisaItaly

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