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Optimality and Stability in Mathematical Programming pp 1–15Cite as

Sufficient conditions for a globally exact penalty function without convexity

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Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 19))

Abstract

In this paper, we consider the nonlinear programming problem P to minimize f(x) subject to g i (x)<=0 for i=1, …, m and x∈X. If X is compact and the number of global optimal solutions is finite, under a suitable constraint qualification, we show that a globally exact penalty function exists. Particularly, we establish a one to one correspondence between global optimal solutions to the original problem and global minimizers of the penalty problem for a sufficiently large, but finite, penalty parameter. A lower bound on the penalty parameter is provided in terms of the Kuhn-Tucker Lagrangian multipliers and lower bounds on the functions involved.

This author’s work is supported under USAFOSR contract F49630-79-C0120.

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References

  1. M.S. Bazaraa and J.J. Goode, “Necessary optimality criteria in mathematical programming in the presence of differentiability”, Journal of Mathematical Analysis and Applications 40 (1972) 609–621.

    Article  MATH  MathSciNet  Google Scholar 

  2. M.S. Bararaa and C.M. Shetty, Nonlinear programming: Theory and Algorithms (Wiley, New York, 1979).

    Google Scholar 

  3. D.P. Bertsekas, “Necessary and sufficient conditions for a penalty method to be exact”, Mathematical Programming 9 (1975) 87–99.

    Article  MATH  MathSciNet  Google Scholar 

  4. C. Charalambous, “A lower bound for the controlling parameter of the exact penalty functions”, Mathematical Programming 15 (1978) 278–290.

    Article  MATH  MathSciNet  Google Scholar 

  5. J.P. Evans, F.J. Gould and J.W. Tolle, “Exact penalty functions in nonlinear programming”, Mathematical Programming 4 (1973) 72–97.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Fletcher, “An exact penalty function for nonlinear programming with inequalities”, Mathematical Programming 5 (1973) 129–150.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Guignard, “Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space”, SIAM Journal on Control 7 (1969) 232–241.

    Article  MATH  MathSciNet  Google Scholar 

  8. S.P. Han, “A globally convergent method for nonlinear programming”, Journal of Optimization Theory and Applications 22 (1977) 297–309.

    Article  MATH  MathSciNet  Google Scholar 

  9. S.P. Han and O.L. Mangasarian, “Exact penalty functions in nonlinear programming”, Mathematical Programming 17 (1979) 251–269.

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Howe, “New conditions for exactness of a simple penalty function”, SIAM Journal on Control 11 (1973) 378–381.

    Article  MATH  MathSciNet  Google Scholar 

  11. O.L. Mangasarian, Nonlinear programming (McGraw-Hill, New York, 1969).

    MATH  Google Scholar 

  12. G.P. McCormick, “An idealized exact penalty function”, in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds., Nonlinear programming, Vol. 3 (Academic Press, New York, 1978) pp. 165–195.

    Google Scholar 

  13. T. Pietrzykowski, “The exact potential method for constrained maxima”, SIAM Journal on Numerical Analysis 6 (1969) 299–304.

    Article  MATH  MathSciNet  Google Scholar 

  14. T. Pietrzykowski, “The potential method for conditioned maxima in locally compact metric spaces”, Numerische Mathematik 14 (1970) 325–329.

    Article  MATH  MathSciNet  Google Scholar 

  15. R.T. Rockafellar, “Theory of subgradients and its applications to problems of optimization”, lecture notes, University of Montreal (1978).

    Google Scholar 

  16. W.I. Zangwill, “Nonlinear programming via penalty functions”, Management Science 13 (1967) 344–358.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    Mokhtar S. Bazaraa

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA

    Jamie J. Goode

Authors
  1. Mokhtar S. Bazaraa
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  2. Jamie J. Goode
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Monique Guignard

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© 1982 The Mathematical Programming Society, Inc.

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Bazaraa, M.S., Goode, J.J. (1982). Sufficient conditions for a globally exact penalty function without convexity. In: Guignard, M. (eds) Optimality and Stability in Mathematical Programming. Mathematical Programming Studies, vol 19. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120980

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  • DOI: https://doi.org/10.1007/BFb0120980

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00849-8

  • Online ISBN: 978-3-642-00850-4

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