Abstract
We study infinite sets of convex functional constraints, with possibly a set constraint, under general background hypotheses which require closed functions and a closed set, but otherwise do not require a Slater point. For example, when the set constraint is not present, only the consistency of the conditions is needed.
We provide hypotheses, which are necessary as well as sufficient, for the overall set of constraints to have the property that there is no gap in Lagrangean duality for every convex objective function defined on ℝn. The sums considered for our Lagrangean dual are those involving only finitely many nonzero multipliers.
In particular, we recover the usual sufficient condition when only finitely many functional constraints are present. We show that a certain compactness condition in function space plays the role of finiteness, when there are an infinite number of functional constraints.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C.E. Blair, “A note on infinite systems of linear inequalities in ℝn”,Journal of Mathematical Analysis and Applications 2 (1974) 150–154.
C.E. Blair, “Convex optimization and Lagrange multipliers”,Mathematical Programming 15 (1978) 87–91.
C.E. Blair, J.M. Borwein and R.G. Jeroslow, “Convex programs and their closures”, GSIA, Carnegie-Mellon University, (Pittsburgh, PA, 1978).
J. Borwein, “Direct theorems in semi-infinite convex programming”, Department of Mathematics, Carnegie-Mellon University, (Pittsburgh, PA, 1980).
A. Charnes, W. W. Cooper and K. O. Kortanek, “On representations of semi-infinite programs which have no duality gaps”,Management Science 12 (1965) 113–121.
R.J. Duffin and L.A. Karlovitz, “An infinite linear program with a duality gap”,Management Science 12 (1965) 122–134.
R.J. Duffin and L.A. Karlovitz, Unpublished note (1971).
R. J. Duffin, L. A. Karlovitz and R. G. Jeroslow, “Duality in semi-infinite linear programming”, Carnegie-Mellon University and Georgia Institute of Technology (Atlanta, GA, 1981).
R.J. Duffin and R.G. Jeroslow, “Lagrangean functions and affine minorants”,Mathematical Programming Study 14 (1981) 48–60.
R.G. Jeroslow, “A limiting Lagrangean for infinitely constrained convex optimization in ℝn”,Journal of Optimization Theory and its Applications 33 (1981) 479–495.
R.G. Jeroslow, “Uniform duality in semi-infinite convex optimization”, MS-81-6, College of Management, Georgia Institute of Technology (Atlanta, GA, 1981).
D.F. Karney, “Duality gaps in semi-infinite linear programming—an approximation problem”,Mathematical Programming 20 (1981) 129–143.
J.L. Kelley and I. Namioka,Linear topological spaces (Springer, New York, 1963).
V. Klee, “The critical set of a convex body”,American Journal of Mathematics 75 (1953) 178–188.
K.O. Kortanek, “Perfect duality in generalized convex programming in finite dimensions”, Department of Mathematics, Carnegie-Mellon University Tech. Report No. 26 (Pittsburgh, PA, 1975).
K.S. Kretchmer, “Programmes in paired spaces”,Canadian Journal of Mathematics 13 (1959) 221–238.
R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, New Jersey, 1970).
J. Stoer and C. Witzgall,Convexity and optimization in finite dimensions: I (Springer, New York, 1970).
Author information
Authors and Affiliations
Additional information
The author's research has been partially supported by Grant ECS8001763 of the National Science Foundation.
Rights and permissions
About this article
Cite this article
Jeroslow, R.G. Uniform duality in semi-infinite convex optimization. Mathematical Programming 27, 144–154 (1983). https://doi.org/10.1007/BF02591942
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02591942