Advertisement

Mathematical Programming

, Volume 47, Issue 1–3, pp 117–141 | Cite as

A bifurcation analysis of the nonlinear parametric programming problem

  • C. A. Tiahrt
  • A. B. Poore
Article

Abstract

The structure of solutions to the nonlinear parametric programming problem with a one dimensional parameter is analyzed in terms of the bifurcation behavior of the curves of critical points and the persistence of minima along these curves. Changes in the structure of the solution occur at singularities of a nonlinear system of equations motivated by the Fritz John first-order necessary conditions. It has been shown that these singularities may be completely partitioned into seven distinct classes based upon the violation of one or more of the following: a complementarity condition, a constraint qualification, and the nonsingularity of the Hessian of the Lagrangian on a tangent space. To apply classical bifurcation techniques to these singularities, a further subdivision of each case is necessary. The structure of curves of critical points near singularities of lowest (zero) codimension within each case is analyzed, as well as the persistence of minima along curves emanating from these singularities. Bifurcation behavior is also investigated or discussed for many of the subcases giving rise to a codimension one singularity.

Key words

Bifurcation singularity parametric programming stability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer,Nonlinear Parametric Optimization (Birkhauser, Basel, 1983).Google Scholar
  2. [2]
    S.-N. Chow and J.K. Hale,Methods of Bifurcation Theory (Springer, New York, 1982).Google Scholar
  3. [3]
    M.G. Crandal and P.H. Rabinowitz, “Bifurcation from simple eigenvalues,”Journal of Functional Analysis 8 (1971) 321–340.Google Scholar
  4. [4]
    A.V. Fiacco,Introduction to Sensitivity and Stability Analysis in Nonlinear Programming (Academic Press, New York, 1983).Google Scholar
  5. [5]
    A.V. Fiacco, ed.,Sensitivity, Stability and Parametric Analysis, Mathematical Programming Study 21 (North Holland, Amsterdam, 1984).Google Scholar
  6. [6]
    G.H. Golub and C.F. Van Loan,Matrix Computations (The Johns Hopkins University Press, Baltimore, MD, 1983).Google Scholar
  7. [7]
    M. Golubitsky and D.G. Schaeffer,Singularities and Groups in Bifurcation Theory, Vol. 1 (Springer, New York, 1985).Google Scholar
  8. [8]
    C.D. Ha, “Application of degree theory in stability of the complementarity problem,” to appear in:Mathematics of Operations Research. Google Scholar
  9. [9]
    G. Iooss and D.D. Joseph,Elementary Stability and Bifurcation Theory (Springer, New York, 1980).Google Scholar
  10. [10]
    H.Th. Jongen, P. Jonker and F. Twilt, “On one-parameter families of sets defined by (in)equality constraints,”Nieuw Archief Voor Wiskunde 3 (1982) 307–322.Google Scholar
  11. [11]
    H.Th. Jongen, P. Jonker and F. Twilt,Nonlinear Optimization in R n:1. Morse Theory, Chebyshev Approximation (Peter Lang, New York, 1983).Google Scholar
  12. [12]
    H.Th. Jongen, P. Jonker and F. Twilt, “Critical sets in parametric optimization,”Mathematical Programming 34 (1984) 333–353.Google Scholar
  13. [13]
    H.Th. Jongen, P. Jonker and F. Twilt, “One-parameter families of optimization problems: Equality constraints,”Journal of Optimization Theory and Applications 48 (1986) 141–161.Google Scholar
  14. [14]
    T. Kato,Perturbation Theory for Linear Operators (Springer, New York, 1966).Google Scholar
  15. [15]
    H.B. Keller, “Numerical solution of bifurcation and nonlinear eigenvalue problems,” in: P.H. Rabinowitz, ed.,Applications of Bifurcation Theory (Academic Press, New York, 1977) pp. 359–384.Google Scholar
  16. [16]
    M. Kojima, “Strongly stable solutions in nonlinear programs,” in: S.M. Robinson, ed.,Analysis and Computation of Fixed Points (Academic Press, New York, 1980) pp. 93–138.Google Scholar
  17. [17]
    M. Kojima and R. Hirabayashi, “Continuous deformation of nonlinear programs,” in:Mathematical Programming Study 21 (1984) 150–198.Google Scholar
  18. [18]
    O.L. Mangasarian and S. Fromovitz, “The Fritz John necessary optimality conditions in the presence of equality and inequality constraints,”Journal of Mathematical Analysis and Applications 17 (1967) 37–47.Google Scholar
  19. [19]
    A.B. Poore and C.A. Tiahrt, “Bifurcation problems in nonlinear parametric programming,”Mathematical Programming 39 (1987) 189–205.Google Scholar
  20. [20]
    S.M. Robinson, “Stability theory for systems of inequalities, part II,”SIAM Journal on Numerical Analysis 13 (1976) 497–513.Google Scholar
  21. [21]
    S.M. Robinson, “Strongly regular generalized equations,”Mathematics of Operations Research 5 (1980) 43–62.Google Scholar
  22. [22]
    Y. Sawaragi, H. Nakayama and T. Tanino,Theory of Multiobjective Optimization (Academic Press, New York, 1985).Google Scholar
  23. [23]
    S. Schecter, “Structure of the first-order solution set of a class of nonlinear programs with parameters,”Mathematical Programming 34 (1986) 84–110.Google Scholar
  24. [24]
    D. Seirsma, “Singularities of functions on boundaries, corners, etc.,”Quarterly Journal of Mathematics, Oxford 32 (1981) 119–127.Google Scholar
  25. [25]
    C.A. Tiahrt, “Nonlinear parametric programming: Critical point bifurcation and persistence of minima,” Ph.D. Thesis, Colorado State University (Fort Collins, CO, 1986).Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1990

Authors and Affiliations

  • C. A. Tiahrt
    • 1
  • A. B. Poore
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of NebraskaLincolnUSA
  2. 2.Department of MathematicsColorado State UniversityFort CollinsUSA

Personalised recommendations