Advertisement

Connected level sets, minimizing sets, and uniqueness in optimization

  • D. H. Martin
Contributed Papers

Abstract

Intimate relationships are investigated between connectedness properties of the lower level sets of a real functionf on a topological spaceX and the uniqueness of suitably defined minimizing sets forf. Two distinct theories are presented, the simpler one pertaining to the LE-level sets
$$LE_\alpha (f) = \{ x \in X|f(x) \leqslant \alpha \} $$
and the other to the LT-level sets
$$LT_\alpha (f) = \{ x \in X|f(x) \leqslant \alpha \} .$$
In each theory, a specific notion of minimizing set is defined in such a way that a functionf having connected level sets can have at most one minimizing set. That this uniqueness is not trivial, however, is shown by the converse result that, ifX is Hausdorff and the sets LEα(f) are all compact, then, in each theory,f has a unique minimizing set only if it has connected level sets. The paper concludes by showing that functions with connected LT-level sets arise naturally in parametric linear programming.

Key Words

Level sets connectedness uniqueness of minimizers parametric programming optimization theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ortega, J. H., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.Google Scholar
  2. 2.
    Avriel, M., andZang, I.,Generalized Arcwise-Connected Functions and Characterization of Local-Global Minimum Properties, Tel-Aviv University, Tel-Aviv, Israel, Working Paper No. 502/77, 1977.Google Scholar
  3. 3.
    Martin, D. H.,Connected Level Sets, Minimizing Sets, and Uniqueness in Optimization, Parts 1 and 2, CSIR, Pretoria, South Africa, Special Reports Nos. WISK-276 and WISK-277, 1977.Google Scholar
  4. 4.
    Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.Google Scholar
  5. 5.
    Polyak, B. T.,Existence Theorems and Convergence of Minimizing Sequences in Extremum Problems, Soviet Mathematics Doklady, Vol. 116, pp. 72–74, 1966.Google Scholar
  6. 6.
    Martin, D. H.,Some Function Classes Closed under Infimal Convolution, Journal of Optimization Theory and Applications, Vol. 25, pp. 597–584, 1978.Google Scholar
  7. 7.
    Dantzig, G. B., Folkman, J., andShapiro, N.,On the Continuity of the Minimum Set of a Continuous Function, Journal of Mathematical Analysis and Applications, Vol. 17, pp. 519–548, 1976.Google Scholar
  8. 8.
    Hall, D. W., andSpencer, G. L. II,Elementary Topology, John Wiley and Sons, New York, New York, 1955.Google Scholar
  9. 9.
    Barnett, S.,Introduction to Mathematical Control Theory, Oxford University Press, Oxford, England, 1975.Google Scholar
  10. 10.
    Martos, B.,Subdefinite Matrices and Quadratic Forms, SIAM Journal on Applied Mathematics, Vol. 17, pp. 1215–1223, 1969.Google Scholar
  11. 11.
    Martos, B.,Quadratic Programming with a Quasiconvex Objective Function, Operations Research, Vol. 19, pp. 87–97, 1971.Google Scholar
  12. 12.
    Cottle, R. W., andFerland, J. A.,Matrix-Theoretic Criteria for the Quasi-Convexity and Pseudo-Convexity of Quadratic Functions, Linear Algebra and Its Applications, Vol. 5, pp. 123–136, 1972.Google Scholar
  13. 13.
    Perold, A.,A Generalization of the Frank-Wolfe Theorem, Mathematical Programming, Vol. 18, pp. 215–227, 1980.Google Scholar
  14. 14.
    Dantzig, G. B.,Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey, 1963.Google Scholar

Copyright information

© Plenum Publishing Corporation 1982

Authors and Affiliations

  • D. H. Martin
    • 1
  1. 1.National Research Institute for Mathematical Sciences, CSIRPretoriaSouth Africa

Personalised recommendations