Skip to main content

Some applications of penalty functions in mathematical programming

Part of the Lecture Notes in Mathematics book series (LNM,volume 1190)

Abstract

By using an exterior penalty function and recent boundedness and existence results for monotone complementarity problems, we give existence and boundedness results, for a pair of dual convex programs, of the following nature. If there exists a point which is feasible for the primal problem and which is interior to the constraints of the Wolfe dual, then the primal problem has a solution which is easily bounded in terms of the feasible point. Furthermore there exists no duality gap. We also show that by solving an exterior penalty problem for only two values of the penalty parameter we obtain an optimal point which is approximately feasible to any desired preassigned tolerance. This result is then employed to obtain an estimate of the perturbation parameter for a linear program which allows us to solve the linear program to any preassigned accuracy by an iterative scheme such as a successive over-relaxation (SOR) method.

Keywords

  • Penalty Function
  • Penalty Parameter
  • Primal Problem
  • Unbounded Sequence
  • Boundedness Result

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Sponsored by the United States Army under Contract No.DAAG29-80-C-0041. This material is based on work sponsored by National Science Foundation Grant MCS-8200632.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.S. Bazaraa and C.M. Shetty. ‘Nonlinear programming theory and algorithms'. Wiley, New York (1979).

    MATH  Google Scholar 

  2. E.J. Beltrami. ‘An algorithmic approach to nonlinear analysis and optimization'. Academic Press, New York (1970).

    MATH  Google Scholar 

  3. A.V. Fiacco and G.P. McCormick. ‘Nonlinear programming: Sequential unconstrained minimization techniques'. Wiley, New York (1968).

    MATH  Google Scholar 

  4. A.M. Geoffrion. ‘Duality in nonlinear programming: a simplified applications-oriented development'. SIAM Review 13, (1971) 1–37.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. O.L. Mangasarian. ‘Nonlinear programming'. McGraw-Hill, New York (1969).

    MATH  Google Scholar 

  6. O.L. Mangasarian. ‘Iterative solution of linear programs'.SIAM Journal of Numerical Analysis 18, (1981) 606–614.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. O.L. Mangasarian. ‘Sparsity-preserving SOR algorithms for separable quadratic and linear programming problems'. Computers and Operations Research 11, (1984) 105–112.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. O.L. Managasarian. ‘Normal solutions of linear programs',Mathematical Programming Study 22,(1984) 206–216.

    CrossRef  MathSciNet  Google Scholar 

  9. O.L. Managasarian. ‘A condition number for differentiable convex inequalities', Mathematics of Operations Research 10, (1985)175–179.

    CrossRef  MathSciNet  Google Scholar 

  10. O.L. Managasarian and L. McLinden. ‘Simple bounds for solutions of monotone complementarity problems and convex programs', Mathematical Programming 32, (1985) 32–40.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. G.P. McCormick. ‘Nonlinear programming theory, algorithms and applications'. Wiley, New York (1983).

    MATH  Google Scholar 

  12. P. Wolfe. ‘A duality theorem for nonlinear programming'.Quarterly of Applied Mathematics 19 (1961) 239–244.

    MathSciNet  MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1986 Springer-Verlag

About this paper

Cite this paper

Mangasarian, O.L. (1986). Some applications of penalty functions in mathematical programming. In: Conti, R., De Giorgi, E., Giannessi, F. (eds) Optimization and Related Fields. Lecture Notes in Mathematics, vol 1190. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076712

Download citation

  • DOI: https://doi.org/10.1007/BFb0076712

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16476-0

  • Online ISBN: 978-3-540-39817-2

  • eBook Packages: Springer Book Archive