Skip to main content
SpringerLink
Log in

Unconstrained and constrained global optimization of polynomial functions in one variable

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In Floudas and Visweswaran (1990), a new global optimization algorithm (GOP) was proposed for solving constrained nonconvex problems involving quadratic and polynomial functions in the objective function and/or constraints. In this paper, the application of this algorithm to the special case of polynomial functions of one variable is discussed. The special nature of polynomial functions enables considerable simplification of the GOP algorithm. The primal problem is shown to reduce to a simple function evaluation, while the relaxed dual problem is equivalent to the simultaneous solution of two linear equations in two variables. In addition, the one-to-one correspondence between the x and y variables in the problem enables the iterative improvement of the bounds used in the relaxed dual problem. The simplified approach is illustrated through a simple example that shows the significant improvement in the underestimating function obtained from the application of the modified algorithm. The application of the algorithm to several unconstrained and constrained polynomial function problems is demonstrated.

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. Dixon, L. C. W. (1990), On Finding the Global Minimum of a Function of One Variable, Presented at the SIAM National Meeting, Chicago.

  2. Dixon, L. C. W. and Szegö, G. P. (1975), Towards Global Optimization, North Holland, Amsterdam.

    Google Scholar 

  3. Evtushenko, Y. G. (1971), Numerical Methods for Finding Global Extrema (Case of Nonuniform Mesh), USSR Computational Mathematics and Mathematical Physics 11, 1390.

    Google Scholar 

  4. Floudas, C. A. and Visweswaran, V. (1990), A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs—I. Theory, Computers and Chemical Engineering 14, 1397.

    Google Scholar 

  5. Floudas, C. A. and Visweswaran, V. (1992), A Primal-Relaxed Dual Global Optimization Approach.

  6. Galperin, E. A. (1985), The Cubic Algorithm, Journal of Mathematical Analysis and Applications 112, 635.

    Google Scholar 

  7. Galperin, E. A. (1987), The Beta-Algorithm, Journal of Mathematical Analysis and Applications 126, 455.

    Google Scholar 

  8. Goldstein, A. A. and Price, J. F. (1971), On Descent From Local Minima, Mathematics of Computation 25, 569.

    Google Scholar 

  9. Hansen, E. R. (1979), Global Optimization Using Interval Analysis: The One-Dimensional Case, Journal of Optimization Theory and Applications 29, 331.

    Google Scholar 

  10. Hansen, P., Jaumard, B., and Lu, S-H. (1991), Global Optimization of Univariate Lipschitz Functions: I. Survey and Properties, Mathematical Programming.

  11. Hansen, P., Jaumard, B., and Lu, S-H. (1991), Global Optimization of Univariate Lipschitz Functions: II. New Algorithms and Computational Comparisons, Mathematical Programming.

  12. Hansen, P., Lu, S-H., and Jaumard, B. (1989), Global Minimization of Univariate Functions by Sequential Polynomial Approximation, International Journal of Computer Mathematics 28, 183.

    Google Scholar 

  13. Moore, R. (1979), Methods and Applications in Interval Analysis, SIAM Studies in Applied Mathematics, Philadelphia.

  14. Piyavskii, S. A. (1972), An Algorithm for Finding the Absolute Extremeum of a Function, USSR Comput. Math. Phys. 12, 57.

    Google Scholar 

  15. Ratschek, H. and Rokne, J. (1988), New Computer Methods for Global Optimization, Halsted Press.

  16. Shen, Z. and Zhu, Y. (1987), An Interval Version of Schubert's Iterative Method for the Localization of the Global Maximum, Computing 38, 275.

    Google Scholar 

  17. Shirov, V. S. (1985), Search for the Global Extremum of a Polynomial on a Parallelopiped, USSR Comput. Maths. Math. Phys. 25, 105.

    Google Scholar 

  18. Timonov, L. N. (1977), An Algorithm for Search of a Global Extremum, Engineering Cybernetics 15, 38.

    Google Scholar 

  19. Visweswaran, V. and Floudas, C. A. (1990a), A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs—II. Application of Theory and Test Problems, Computers and Chemical Engineering 14, 1419.

    Google Scholar 

  20. Visweswaran, V. and Floudas, C. A. (1990b), New Properties and Computational Improvement of the GOP Algorithm for Problems with Quadratic Objective Function and Constraints.

  21. Wilkinson, J. H. (1963), Rounding Errors in Algebraic Processes, Prentice-Hill, Englwood Cliffs, N.J.

    Google Scholar 

  22. Wingo, D. R. (1985), Globally Minimizing Polynomials without Evaluating Derivatives, International Journal of Computer Mathematics 17, 287.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chemical Engineering Department, Princeton University, 08544-5263, Princeton, N.J., U.S.A.

    V. Visweswaran & C. A. Floudas

Authors
  1. V. Visweswaran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. A. Floudas
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Visweswaran, V., Floudas, C.A. Unconstrained and constrained global optimization of polynomial functions in one variable. J Glob Optim 2, 73–99 (1992). https://doi.org/10.1007/BF00121303

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00121303

Key words

Search

Navigation