Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Secant approximation methods for convex optimization

  • Chapter
  • First Online:
Mathematical Programming at Oberwolfach

Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 14))

Abstract

The methods discussed are based on local piecewise-linear secant approximations to continuous convex objective functions. Such approximations are easily constructed and require only function evaluations rather than derivatives. Several related iterative procedures are considered for the minimization of separable objectives over bounded closed convex sets. Computationally, the piecewise-linear approximation of the objective is helpful in the case that the original problem has only linear constraints, since the subproblems in this case will be linear programs. At each iteration, upper and lower bounds on the optimal value are derived from the piecewise-linear approximations. Convergence to the optimal value of the given problem is established under mild hypotheses. The method has been successfully tested on a variety of problems, including a water supply problem with more than 900 variables and 600 constraints.

Research supported by National Science Foundation Grant MCS74-20584 A02.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Avriel, Nonlinear programming analysis and methods (Prentice-Hall, Englewood Cliffs, NJ, 1976).

    MATH  Google Scholar 

  2. A. Bachem and B. Korte, “Quadratic programming over transportation polytopes”, Report 7767-OR, Institut für ökonometrie und Operations Research Bonn, 1977.

    Google Scholar 

  3. E.M.L. Beale, Mathematical programming in practice (Sir Isaac Pitman & Sons, London, 1968).

    Google Scholar 

  4. A. Charnes and C. E. Lemke, “Minimization of nonlinear separable convex functionals”, Naval Research Logistics Quarterly 1 (1954) 301–312.

    Article  MathSciNet  Google Scholar 

  5. M. Collins, L. Cooper, R. Helgason, J. Kennington and L. LeBlanc, “Solving the pipe network analysis problem using optimization techniques”, Management Science 24 (1978) 747–760.

    Article  MATH  MathSciNet  Google Scholar 

  6. G.B. Dantzig, Linear programming and extensions (Princeton University Press, Princeton, NJ, 1963).

    MATH  Google Scholar 

  7. G.B. Dantzig, J. Folkman and N. Shapiro, “On the continuity of the minimum set of a continuous function”, Journal of Mathematical Analysis and Applications 17 (1967) 519–548.

    Article  MATH  MathSciNet  Google Scholar 

  8. G.B. Dantzig, S. Johnson and W. White, “A linear programming approach to the chemical equilibrium problem,” Management Science 5 (1958) 38–43.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. Fletcher, “An algorithm for solving linearly constrained optimization problems”, Mathematical Programming 2 (1972) 133–165.

    Article  MATH  MathSciNet  Google Scholar 

  10. A.M. Geoffrion, “Objective function approximations in mathematical programming”, Mathematical Programming 13, (1977) 23–27.

    Article  MATH  MathSciNet  Google Scholar 

  11. R.E. Griffith and R.A. Stewart, “A nonlinear programming technique for the optimization of continuous processing systems”, Management Science 7 (1961) 379–392.

    Article  MATH  MathSciNet  Google Scholar 

  12. O. Gross, Class of discrete type minimization problem, RM-1644, (Rand Corporation, Santa Monica, CA, 1956).

    Google Scholar 

  13. G. Hadley, Nonlinear and dynamic programming (Addison-Wesley, Reading, MA, 1964).

    MATH  Google Scholar 

  14. R.E. Marsten, W.W. Hogan and J.W. Blankenship, “The boxstep method for large-scale optimization”, Operations Research 23 (1975) 389–405.

    Article  MATH  MathSciNet  Google Scholar 

  15. R.R. Meyer, “A class of nonlinear integer programs solvable by a single linear program”, SIAM Journal on Control and Optimization 15 (1977) 935–946.

    Article  MATH  MathSciNet  Google Scholar 

  16. R.R. Meyer, “Two-segment separable programming”, Management Science 25 (1979) 385–395.

    Article  MATH  MathSciNet  Google Scholar 

  17. R.R. Meyer, “Computational aspects of two-segment separable programming”, in preparation (1980).

    Google Scholar 

  18. R.R. Meyer and M.L. Smith, “Algorithms for a class of ‘convex’ nonlinear integer programs”, in: W.W. White, ed., Computers and mathematical programming NBS special publication 502 (1978).

    Google Scholar 

  19. C.E. Miller, “The simplex method for local separable programming”, in: R.L. Graves and P. Wolfe, eds., Recent advances in mathematical programming (McGraw-Hill, New York, 1963).

    Google Scholar 

  20. H. Müller-Merbach, “Die Methode der ‘direkten Koeffizientanpassung’ (μ-Form) des Separable Programming”, Unternehmensforschung 14 (3) (1970) 197–214.

    Article  MATH  MathSciNet  Google Scholar 

  21. R.T. Rockafellar, “Optimization in networks”, lecture notes, University of Washington, Seattle, (1976).

    Google Scholar 

  22. T.L. Saaty, Optimization in integers and related extremal problems (McGraw-Hill, New York. 1970).

    MATH  Google Scholar 

  23. G.H. Symonds, “Chance-constrained equivalents of some stochastic program”, Operations Research 15 (1967) 495–512.

    Article  MATH  MathSciNet  Google Scholar 

  24. J.Z. Teng, “Exact distribution of the Kruskal-Wallis H test and the asymptotic efficiency of the Wilcoxon test with ties”, Ph.D. thesis, University of Wisconsin-Madison, Madison (1978).

    Google Scholar 

  25. L.S. Thakur, “Error analysis for convex separable programs: the piecewise linear approximation and the bounds on the optimal objective value”, SIAM Journal on Applied Mathematics (1978) 704–714.

    Google Scholar 

  26. H.M. Wagner, Principles of operational research with applications to managerial decisions (Prentice-Hall, Englewood Cliffs, NJ, 1969).

    Google Scholar 

  27. R. Wets, “Programming under uncertainty: the equivalent convex program,” Siam Journal on Applied Mathematics 14 (1966) 89–105.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Wisconsin, 53706, Madison, WI, U.S.A.

    C. Y. Kao & R. R. Meyer

Authors
  1. C. Y. Kao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. R. Meyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

H. König B. Korte K. Ritter

Rights and permissions

Reprints and permissions

Copyright information

© 1981 The Mathematical Programming Society

About this chapter

Cite this chapter

Kao, C.Y., Meyer, R.R. (1981). Secant approximation methods for convex optimization. In: König, H., Korte, B., Ritter, K. (eds) Mathematical Programming at Oberwolfach. Mathematical Programming Studies, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120926

Download citation

  • DOI: https://doi.org/10.1007/BFb0120926

  • Received:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00805-4

  • Online ISBN: 978-3-642-00806-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation