Abstract
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.
Similar content being viewed by others
References
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.
J.M. Borwein, “Proper efficient points for maximizations with respect to cones”,SIAM Journal on Control and Optimization 15 (1977) 57–63.
J.M. Borwein, “The limiting Lagrangean as a consequence of Helly's theorem”, Dalhousie Research Report 7 (1979) (To appear inJ.O.T.A.).
J.M. Borwein and H. Wolkowicz, “Convex programming without constraint qualification” (To appear inMathematical Programming).
N. Bourbaki,Elément de mathematique: Livre III, Topologie Generale (Paris 1948).
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.
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.
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.
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.
R.J. Duffin and L.A. Karlovitz, “An infinite linear program with duality gap”,Management Science 12 (1965) 122–134.
A.M. Geoffrion, “Proper efficiency and the theory of vector maximization”,Journal of Mathematical Analysis and Applications 22 (1963) 618–630.
J.L. Kelley,General topology (Princenton University Press, Princeton, NJ, 1953).
V. Klee, “The critical set of a convex body”,American Journal of Mathematics 75 (1953) 178–188.
D.G. Luenberger,Optimization by vector space methods (John Wiley, New York, 1969).
D.G. Luenberger, “Quasi-convex programming”,SIAM Journal of Applied Mathematics 16 (1968) 1090–1095.
L. McLinden, “Affine minorants minimizing the sum of convex functions”,Journal of Optimization Theory and Applications 24 (1978) 569–583.
J. Ponstein, “Seven kinds of convexity”,SIAM Review 2 (1967) 115–119.
B.N. Pshenichnyi,Necessary conditions for an extremum (Marcel Dekker, New York 1971).
R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton NJ, 1970).
R.T. Rockafellar, “Helly's theorem and minima of convex functions”,Duke Mathematical Journal 32 (1965) 381–397.
J. Stoer and C. Witzgall,Convexity and optimization in finite dimensions, I (Springer, Berlin, 1970).
A. Wilansky,Modern methods in topological vector spaces (McGraw-Hill, New York, 1978).
Author information
Authors and Affiliations
Additional information
Research partially funded by NSERC A4493.
Rights and permissions
About this article
Cite this article
Borwein, J.M. Direct theorems in semi-infinite convex programming. Mathematical Programming 21, 301–318 (1981). https://doi.org/10.1007/BF01584251
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01584251