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Journal of Convex Analysis 08 (2001), No. 2, 423--446
Copyright Heldermann Verlag 2001

The Conjugates, Compositions and Marginals of Convex Functions

S. P. Fitzpatrick
Dept. of Mathematics and Statistics, University of Western Australia, Nedlands 6907, Australia

S. Simons
Dept. of Mathematics, University of California, Santa Barbara, CA 96106-3080, U.S.A.

Continuing the work of Hiriart-Urruty and Phelps, we discuss (in both locally convex spaces and Banach spaces) the formulas for the conjugates and subdifferentials of the precomposition of a convex function by a continuous linear mapping and the marginal function of a convex function by a continuous linear mapping. We exhibit a certain (incomplete) duality between the operations of precomposition and marginalization. Our results lead easily to Thibault's proof of the maximal monotonicity of the subdifferential of a proper, convex lower semicontinuous function on a Banach space. We show that some of the Hiriart-Urruty-Phelps results on ε-subdifferentials have analogs in terms of the "ε-enlargement" of the subdifferential. We obtain new results on the conjugates and subdifferentials of sums of convex functions without constraint qualifications and also of episums of convex functions. We discuss constrained minimization on non-closed convex subsets of a Banach space.

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