Abstract
In Ref. 1, Soyster has given a rather complicated proof of the absence of a duality gap, under a certain interiority condition, for a variant of a pair of optimization problems introduced by Ben-Israel, Charnes, and Kortanek (Ref. 2). A proof can be given directly (and under weaker conditions) by a simple application of a Lagrange multiplier theorem on convex programming in abstract spaces (Ref. 3).
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References
Soyster, A. L.,A Note on Duality Gaps in Linear Programming over Convex Sets, Journal of Optimization Theory and Applications, Vol. 13, No. 4, 1974.
Ben-Israel, A., Charnes, A., andKortanek, K. O.,Asymptotic Duality over Closed Convex Sets, Journal of Mathematical Analysis and Applications, Vol. 35, No. 3, 1971.
Luenberger, D. G.,Optimization by Vector Space Methods, John Wiley and Sons, New York, New York, 1969.
Choquet, G.,Lectures on Analysis, Vol. 2, W. A. Benjamin, Inc., New York, New York, 1969.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Ben-Israel, A., Charnes, A., andKortanek, K. O.,Duality and Asymptotic Solvability over Cones, Bulletin of the American Mathematical Society, Vol. 75, No. 2, 1969.
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Communicated by A. Miele
This research was partially supported by the National Science Foundation, Grant No. GP-31091X2.
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Balder, E.J. Comment on a note on duality gaps in linear programming over convex sets. J Optim Theory Appl 17, 343–346 (1975). https://doi.org/10.1007/BF00933884
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DOI: https://doi.org/10.1007/BF00933884