Skip to main content
SpringerLink
Log in

A finite steepest-ascent algorithm for maximizing piecewise-linear concave functions

  • Technical Note
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this note, we describe a finitely convergent steepest-ascent scheme for maximizing piecewise-linear concave functions. Given any point, the algorithm moves along the direction of steepest ascent, that is, along the shortest subgradient, until a new ridge is reached. The overall process is then repeated by moving along the new steepest-ascent direction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Demyanov, V. F.,Algorithms for Some Minimax Problems, Journal of Computer and Systems Sciences, Vol. 2, pp. 342–380, 1968.

    Google Scholar 

  2. Bazaraa, M. S., andGoode, J. J.,A Survey of Various Tactics for Generating Lagrangian Multipliers in the Context of Lagrangian Duality, Georgia Institute of Technology, School of Industrial and Systems Engineering, Working Paper, 1974.

  3. Fisher, M. L., Northup, W. D., andShapiro, J. F.,Using Duality to Solve Discrete Optimization Problems: Theory and Computational Experience, Mathematical Programming Study, Vol. 3, pp. 56–94, 1975.

    Google Scholar 

  4. Lasdon, L. S.,Optimization Theory for Large Systems, The MacMillan Company, New York, New York, 1970.

    Google Scholar 

  5. Geoffrion, A. M.,Elements of Large Scale Mathematical Programming, Parts I and II, Management Science, Vol. 16, pp. 652–691, 1970.

    Google Scholar 

  6. Wolfe, P.,Algorithm for a Least Distance Programming Problem, Mathematical Programming Study, Vol. 3, pp. 145–173, 1975.

    Google Scholar 

  7. Grinold, R. C.,Steepest Ascent for Large Scale Linear Programs, SIAM Review, Vol. 14, pp. 447–464, 1972.

    Google Scholar 

  8. Dantzig, G. B., andWolfe, P.,The Decomposition Algorithm for Linear Programming, Econometrica, Vol. 29, pp. 767–778, 1961.

    Google Scholar 

  9. Brooks, R., andGeoffrion, A. M.,Finding Everett's Lagrange Multipliers by Linear Programming, Operations Research, Vol. 14, pp. 1149–1153, 1966.

    Google Scholar 

  10. Held, M., Wolfe, P., andGrowder, H. D.,Validation of Subgradient Optimization, Mathematical Programming, Vol. 6, pp. 62–88, 1974.

    Google Scholar 

  11. Eaves, C. B.,Solving Piecewise Linear Convex Equations, Mathematical Programming Study, Vol. 1, pp. 96–119, 1974.

    Google Scholar 

  12. Eaves, C. B., andScarf, H.,The Solution of Systems of Piecewise Linear Equations, Mathematics of Operations Research, Vol. 1, pp. 1–31, 1976.

    Google Scholar 

  13. Tucker, A. W.,Least-Distance Programming, Proceedings of the Princeton Symposium on Mathematical Programming, Edited by H. W. Kuhn, Princeton University Press, Princeton, New Jersey, 1971.

    Google Scholar 

  14. Bazaraa, M. S., Goode, J. J., andRardin, R. L.,An Algorithm for Finding the Shortest Element of a Polyhedral Set with Applications to Lagrangian Duality, Journal of Mathematical Analysis and Applications (to appear).

  15. Luenberger, D. G.,Optimization by Vector Space Methods, The MacMillan Company, New York, New York, 1969.

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    M. S. Bazaraa (Associate Professor) & R. L. Rardin (Assistant Professor)

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

    J. J. Goode (Associate Professor)

Authors
  1. M. S. Bazaraa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. J. Goode
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. L. Rardin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by M. Avriel

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bazaraa, M.S., Goode, J.J. & Rardin, R.L. A finite steepest-ascent algorithm for maximizing piecewise-linear concave functions. J Optim Theory Appl 25, 437–442 (1978). https://doi.org/10.1007/BF00932904

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00932904

Key Words

Search

Navigation