Recursive Piecewise-Linear Approximation Methods for Nonlinear Networks

  • R. R. Meyer
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 199)


The use of recursive piecewise-linear approximations in network optimization problems with convex separable objectives allows the utilization of fast solution methods for the corresponding linear network subproblems. Piecewise-linear approximations also enjoy many advantages over other types of approximations, including the ability to utilize simultaneously information from infeasible as well as feasible points (so that results of Lagrangian relaxations may be directly employed in the approximation), guaranteed decrease in the objective without the need for any line search, and easily computed and tight bounds on the optimal value. The speed of this approach will be exemplified by the presentation of computational results for large-scale nonlinear networks arising from econometric and water supply system applications.


Line Search Lagrangian Relaxation Linear Network Nonlinear Integer Program Minimum Cost Network 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bachem, Achim and Korte, Bernhard, “Quadratic Programming over Transportation Polytopes”, Report No 7767-OR, Institute für Okonometrie und Operations Research, Bonn, 1977.Google Scholar
  2. [2]
    Beale, E.M.L., Mathematical Programming in Practice, Sir Isaac Pitman & Sons Ltd., London, 1968.Google Scholar
  3. [3]
    Bradley, Gordon H., Brown, Gerald G., and Graves, Glenn W., “Design and Implementation of Large Scale Primal Transshipment Algorithms”, Man Sci., 24, pp. 1–34, 1977.Google Scholar
  4. [4]
    Collins, M., Cooper, L., Helgason, R., Kennington, J., and LeBlanc, L., “Solving the Pipe Network Analysis Problem Using Optimization Techniques”, Man. Sci., 24, no. 747–760, 1978.Google Scholar
  5. [5]
    Geoffrion, A.M., “Objective Function Approximations for Mathematical Programming”, Mathematical Programming, 13, pp. 23–27, 1977.CrossRefGoogle Scholar
  6. [6]
    Glover, F. and Klingman, D., “Real World Applications of Network Related Problems and Breakthroughs in Solving Them Efficiently”, ACM Transactions on Mathematical Software, 1, pp. 47–55, 1975.CrossRefGoogle Scholar
  7. [7]
    Grigoriadis, Michael D. and Hsu, Tau, “RNET The Rutgers Minimum Cost Network Flow Subroutines”, SIGMAP Bulletin, pp. 17–18, April, 1979.Google Scholar
  8. [8]
    Kao, C.Y. and Meyer, R.R., “Secant Approximation Methods for Convex Optimization”, University of Wisconsin-Madison Computer Sciences Technical Report #352, 1979, to appear in Math. Prog. Study, 14.Google Scholar
  9. [9]
    Meyer, R.R., “A Class of Nonlinear Integer Programs Solvable by a Single Linear Program”, SICOP, 15, pp. 935–946, 1977.Google Scholar
  10. [10]
    Meyer, R.R., “Two-Segment Separable Programming, Man. Sci., 25, pp. 385–395, 1979.Google Scholar
  11. [11]
    Meyer, R.R., “Computational Aspects of Two-Segment Separable Programming”, University of Wisconsin-Madison Computer Sciences Department Technical Report #382, 1980.Google Scholar
  12. [12]
    Meyer, R.R. and Smith, M.L.., “Algorithms for a Class of ‘Convex’ Nonlinear integer Programs”, in Computers and Mathematical Programming, ed. by W.W. White, NBS Special Publication 502, 1973.Google Scholar
  13. [13]
    Muller-Merbach, H., “Die Methode der ‘direkten Koeffizientanpassung’ (p-Form) des Separable Programming”, Unternehmensforschung, Band 14, Heft 3, pp. 197–214, 1970.Google Scholar
  14. [14]
    Rockafellar, R.T., Optimization in Networks, Lecture Notes, University of Washington, 1976.Google Scholar
  15. [15]
    Saaty, T.L., Optimization in Integers and Related Extremal Problems, McGraw-Hill, New York, 1970.Google Scholar
  16. [16]
    Thakur, L.S., “Error Analysis for Convex Separable Programs: The Piecewise Linear Approximation and the Bounds on the Optimal Objective Value”, SIAM J. App. Math., pp. 704–714, 1978.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

  • R. R. Meyer
    • 1
  1. 1.Computer Sciences DepartmentUniversity of WisconsinMadisonUSA

Personalised recommendations