Abstract
We give hypotheses, valid in reflexive Banach spaces (such as L p for ∞>p>1 or Hilbert spaces), for a certain modification of the ordinary lagrangean to close the duality gap, in convex programs with (possibly) infinitely many constraint functions.
Our modification of the ordinary lagrangean is to perturb the criterion function by a linear term, and to take the limit of this perturbed lagrangean as the norm of this term goes to zero.
We also review the recent literature on this topic of the “limiting lagrangean”.
Partially supported by grant DAAG29-80-C00317, Army Research Office, Research Triangle Park, North Carolina, U.S.A.
Partially supported by grant ENG7900284 of the National Science Foundation.
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References
A. Charnes, W.W. Cooper and K.O. Kortanek, “Duality in semi-infinite programs and some works of Haar and caratheodory”, Management Science 9 (1965) 209–229.
C.E. Blair, J. Borwein and R.G. Jeroslow, “Convex programs and their closures”, Management science series, GSIA, Carnegie-Mellon University, and Georgia Institute of Technology (September 1978).
J. Borwein, “The limiting lagrangean as a consequence of helly’s theorem” (November 1978).
J. Borwein, “A note on perfect duality and limiting lagrangeans” (November 1978, revised January 1979).
R.J. Duffin, “Infinite programs”, in: H.W. Kuhn and A.W. Tucker, eds., Linear inequalities and related systems (Princeton University Press, Princeton, NJ, 1956) 157–170.
R.J. Duffin, “Convex analysis treated by linear programming”, Mathematical Programming 4 (1973) 125–143.
R.J. Duffin and R.G. Jeroslow, “The limiting lagrangean in reflexive spaces” (December 1978).
R.J. Duffin and R.G. Jeroslow, “The limiting lagrangean” (April 1979, revised July 1979).
R.J. Duffin and L.A. Karlovitz, “An infinite linear program with a duality gap”, Management Science 12 (1965) 122–134.
R.B. Holmes, Geometric functional analysis and its applications (Springer, New York, 1975).
R.G. Jeroslow, “A limiting lagrangean for infinitely-constrained convex optimization in R n”, Journal of Optimization Theory and Applications, to appear.
J.L. Kelley and I. Namioka, Linear topological spaces (Springer, New York, 1963).
L. McLinden, “Affine minorants minimizing the sum of convex functions”, Journal of Optimization Theory and Applications 24 (April 1978).
R.T. Rockafellar, Convex analysis (Princeton University Press, Princeton, 1970).
R.T. Rockafellar, Conjugate duality and optimization. Conference board of the mathematical sciences 16 (SIAM Publications, Philadelphia, 1974).
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© 1981 The Mathematical Programming Society
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Duffin, R.J., Jeroslow, R.G. (1981). Lagrangean functions and affine minorants. In: König, H., Korte, B., Ritter, K. (eds) Mathematical Programming at Oberwolfach. Mathematical Programming Studies, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120920
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DOI: https://doi.org/10.1007/BFb0120920
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