Abstract
For convex optimization inR n,we show how a minor modification of the usual Lagrangian function (unlike that of the augmented Lagrangians), plus a limiting operation, allows one to close duality gaps even in the absence of a Kuhn-Tucker vector [see the introductory discussion, and see the discussion in Section 4 regarding Eq. (2)]. The cardinality of the convex constraining functions can be arbitrary (finite, countable, or uncountable).
In fact, our main result (Theorem 4.3) reveals much finer detail concerning our limiting Lagrangian. There are affine minorants (for any value 0<θ≤1 of the limiting parameter θ) of the given convex functions, plus an affine form nonpositive onK, for which a general linear inequality holds onR nAfter substantial weakening, this inequality leads to the conclusions of the previous paragraph.
This work is motivated by, and is a direct outgrowth of, research carried out jointly with R. J. Duffin.
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Duffin, R. J., andJeroslow, R. G.,Affine Minorants and Lagrangian Functions, Mathematical Programming (to appear).
Charnes, A., Cooper, W. W., andKortanek, K. O.,On Representations of Semi-Infinite Programs Which Have No Duality Gaps, Management Science, Vol. 12, No. 1, 1965.
Duffin, R. J.,Convex Analysis Treated by Linear Programming, Mathematical Programming, Vol. 4, No. 2, 1973.
Blair, C. E.,Convex Optimization and Lagrange Multipliers, Mathematical Programming, Vol. 15, pp. 87–91, 1978.
McLinden, L.,Affine Minorants Minimizing the Sum of Convex Functions, Journal of Optimization Theory and Applications, Vol. 24, No. 4, 1978.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Blair, C. E., Borwein, J., andJeroslow, R. G.,Convex Programs and Their Closures, 1978 (manuscript).
Blair, C. E.,A Note on Infinite Systems of Linear Inequalities in R n,Journal of Mathematical Analysis and Applications, Vol. 48, No. 1, 1974.
Kortanek, K. O.,Constructing a Perfect Duality in Infinite Programming, Applied Mathematics and Optimization, Vol. 3, No. 4, 1977.
Duffin, R. J., andKarlovitz, L. A.,An Infinite Linear Program with a Duality Gap, Management Science, Vol. 12, No. 1, 1965.
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Communicated by O. L. Mangasarian
This research was supported by NSF Grant No. GP-37510X1 and ONR Contract No. N00014-75-C0621, NR-047-048. This paper was presented at “Constructive Approaches to Mathematical Models,” a symposium in honor of R. J. Duffin, Pittsburgh, Pennsylvania, 1978. The author is grateful to Professor Duffin for discussions relating to the work reported here.
The author wishes to thank R. J. Duffin for reading an earlier version of this paper and making numerous suggestions for improving it, which are incorporated here. Our exposition and proofs have profited from comments of C. E. Blair and J. Borwein.
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Jeroslow, R.G. A limiting Lagrangian for infinitely constrained convex optimization inR n . J Optim Theory Appl 33, 479–495 (1981). https://doi.org/10.1007/BF00935754
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DOI: https://doi.org/10.1007/BF00935754