Conjugacy correspondences: a unified view

McLinden, L.

doi:10.1090/S0002-9947-1975-0365276-8

Conjugacy correspondences: a unified view

Author: L. McLinden
Journal: Trans. Amer. Math. Soc. 203 (1975), 257-274
MSC: Primary 49A25; Secondary 46G99
DOI: https://doi.org/10.1090/S0002-9947-1975-0365276-8
MathSciNet review: 0365276
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Abstract: As preparation for a duality theory for saddle programs, a partial conjugacy correspondence is developed among equivalence classes of saddle functions. Three known conjugacy correspondences, including Fenchel’s correspondence among convex functions and Rockafellar’s extension of it to equivalence classes of saddle functions, are shown to be degenerate special cases. Additionally, two new correspondences are brought to light as further special cases. Various questions are answered concerning the lower and upper closures and effective domain of the resulting equivalence class, as well as the effect the correspondence has on the related purely convex function and the subdifferential mapping.

Arne Brøndsted, Conjugate convex functions in topological vector spaces, Mat.-Fys. Medd. Danske Vid. Selsk. 34 (1964), no. 2, 27 pp. (1964). MR 166580
W. Fenchel, On conjugate convex functions, Canad. J. Math. 1 (1949), 73–77. MR 28365, DOI https://doi.org/10.4153/cjm-1949-007-x

---, Convex cones, sets and functions, mimeographed lecture notes, Princeton University, 1951.

L. McLinden, Dual operations on saddle functions, Trans. Amer. Math. Soc. 179 (1973), 363–381. MR 316097, DOI https://doi.org/10.1090/S0002-9947-1973-0316097-1

---, Minimax problems, saddle functions and duality, Thesis, University of Washington, 1971; revised and expanded as Technical Summary Report #1190, Math. Res. Center, University of Wisconsin, Madison, Wis., 1971.

Jean-Jacques Moreau, Théorèmes “inf-sup”, C. R. Acad. Sci. Paris 258 (1964), 2720–2722 (French). MR 161127

---, Fonctionelles convexes, lecture notes, Séminaire sur “Les équations aux dérivées partielles,” Collège de France, 1966.

R. T. Rockafellar, Minimax theorems and conjugate saddle-functions, Math. Scand. 14 (1964), 151–173. MR 175037, DOI https://doi.org/10.7146/math.scand.a-10714
R. Tyrrell Rockafellar, A general correspondence between dual minimax problems and convex programs, Pacific J. Math. 25 (1968), 597–611. MR 230548

---, Convex analysis, Princeton Math. Ser., no. 28, Princeton Univ. Press, Princeton, N. J., 1970. MR 43 #445.

R. Tyrrell Rockafellar, Saddle-points and convex analysis, Differential Games and Related Topics (Proc. Internat. Summer School, Varenna, 1970) North-Holland, Amsterdam, 1971, pp. 109–127. MR 0285947