Mathematical Programming

, Volume 21, Issue 1, pp 301–318 | Cite as

Direct theorems in semi-infinite convex programming

  • J. M. Borwein


We show that a semi-infinite quasi-convex program with certain regularity conditions possesses finitely constrained subprograms with the same optimal value. This result is applied to various problems.

Key words

Semi-infinite Programs Open Helly-type Theorems Finite Subprograms Convex Programs Multi-criteria Programs Quasi-convex Programs Quasi-differentiable Programs 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Ben-Tal, A. Ben-Israel and E. Rosinger, “A Helly-type theorem and semi-infinite programming”,Constructive approaches to mathematical models (Academic Press, New York, 1979) pp. 127–135.Google Scholar
  2. [2]
    J.M. Borwein, “Proper efficient points for maximizations with respect to cones”,SIAM Journal on Control and Optimization 15 (1977) 57–63.Google Scholar
  3. [3]
    J.M. Borwein, “The limiting Lagrangean as a consequence of Helly's theorem”, Dalhousie Research Report 7 (1979) (To appear inJ.O.T.A.).Google Scholar
  4. [4]
    J.M. Borwein and H. Wolkowicz, “Convex programming without constraint qualification” (To appear inMathematical Programming).Google Scholar
  5. [5]
    N. Bourbaki,Elément de mathematique: Livre III, Topologie Generale (Paris 1948).Google Scholar
  6. [6]
    A. Charnes, W.W. Cooper and K.O. Kortanek, “Duality in semi-infinite programs and some works of Haar and Carathedory”,Management Science 9 (1963) 209–228.Google Scholar
  7. [7]
    A. Charnes, W.W. Cooper and K. Kortanek, “On representations of semi-infinite programs which have no duality gaps”,Management Science 9 (1965) 113–121.Google Scholar
  8. [8]
    A. Charnes, W.W. Cooper and K. Kortanek, “Duality, Haar programs and finite sequence spaces”,Proceeding of the National Academy of Science 48 (1962) 783–786.Google Scholar
  9. [9]
    L. Danzer, B. Grünbaum and V.L. Klee, “Helly's theorem and its relatives”, in: V.L. Klee ed.,Convexity, Proceedings of Symposia in Pure Mathematics, Vol. VII (Am. Math. Soc. Providence, R.I., 1963) 101–180.Google Scholar
  10. [10]
    R.J. Duffin and L.A. Karlovitz, “An infinite linear program with duality gap”,Management Science 12 (1965) 122–134.Google Scholar
  11. [11]
    A.M. Geoffrion, “Proper efficiency and the theory of vector maximization”,Journal of Mathematical Analysis and Applications 22 (1963) 618–630.Google Scholar
  12. [12]
    J.L. Kelley,General topology (Princenton University Press, Princeton, NJ, 1953).Google Scholar
  13. [13]
    V. Klee, “The critical set of a convex body”,American Journal of Mathematics 75 (1953) 178–188.Google Scholar
  14. [14]
    D.G. Luenberger,Optimization by vector space methods (John Wiley, New York, 1969).Google Scholar
  15. [15]
    D.G. Luenberger, “Quasi-convex programming”,SIAM Journal of Applied Mathematics 16 (1968) 1090–1095.Google Scholar
  16. [16]
    L. McLinden, “Affine minorants minimizing the sum of convex functions”,Journal of Optimization Theory and Applications 24 (1978) 569–583.Google Scholar
  17. [17]
    J. Ponstein, “Seven kinds of convexity”,SIAM Review 2 (1967) 115–119.Google Scholar
  18. [18]
    B.N. Pshenichnyi,Necessary conditions for an extremum (Marcel Dekker, New York 1971).Google Scholar
  19. [19]
    R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton NJ, 1970).Google Scholar
  20. [20]
    R.T. Rockafellar, “Helly's theorem and minima of convex functions”,Duke Mathematical Journal 32 (1965) 381–397.Google Scholar
  21. [21]
    J. Stoer and C. Witzgall,Convexity and optimization in finite dimensions, I (Springer, Berlin, 1970).Google Scholar
  22. [22]
    A. Wilansky,Modern methods in topological vector spaces (McGraw-Hill, New York, 1978).Google Scholar

Copyright information

© The Mathematical Programming Society 1981

Authors and Affiliations

  • J. M. Borwein
    • 1
  1. 1.Mathematics DepartmentDalhousie UniversityHalifaxCanada

Personalised recommendations