# Abstract Convex Analysis (Wiley-Interscience and Canadian by Ivan Singer

By Ivan Singer

This publication examines summary convex research and provides the result of fresh examine, particularly on parametrizations of Minkowski kind dualities and of conjugations of sort Lau. It explains the most strategies via instances and particular proofs.

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The extension of the usual multiplication x by the convention -Hoc X 0 = 0 X -koo = +oo. 51) (for (R + \{0 } , x)). We will present this approach briefly in the Notes and Remarks to Chapter 8. 49) means that every conjugation c : Rx R w is a duality. 50)) another "second condition". 56) where v and A stand for (pointwise) sup and inf in R x and R w , respectively. 57) rEX hence the correspondence A —> 1/f between v-dualities and coupling functions is oneto-one. 57) becomes f ( w ) = sup [w(x) A —f (x)} (f E R X , w E X * ).

113) yields various modified Lagrangians. 113) one obtains the augmented Lagrangian of Rockafellar [237]. , with simultaneous inequality constraints u(y) 0, where u : R" RP, and equality constraints v(y) = 0, where y : R" —> R", can be treated similarly. A large number of other examples of modified Lagrangians, introduced and studied by several authors, which fit into the above general framework can be found, for example, in the papers of Balder [19], Dolecki and Kurcyusz [63], and Lindberg [159].

We will present this approach briefly in the Notes and Remarks to Chapter 9. 7 Abstract Subdtfferentials Another main concept of (usual) convex analysis, with applications in optimization theory and other fields, is that of a subdifferential. We recall that if X = R", or, more generally, if X is a locally convex space with conjugate space X*, and if f : X —> R and X0 E X are such that f (xo) E R, the subdifferential of f at xo is the subset af (x0) of X* defined by af (x0) = {WO E X * I WO(X) WO(X0) f (x) — f(x0) (x e X)).