By Ivan Singer

This publication examines summary convex research and provides the result of fresh learn, in particular on parametrizations of Minkowski kind dualities and of conjugations of style Lau. It explains the most ideas via circumstances and distinct proofs.

**Read or Download Abstract Convex Analysis (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts) PDF**

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28) yields that —oc is convex with respect to W. It is obvious (by taking W' to be a singleton {w}) that every w E W is W-convex. In the case where X is a compact topological space, W is a cone of continuous functions on X, and f is continuous, the above concept of a W-convex function is due to Kutateladze and Rubinov [151], [152]. By the discussion above, the functions f -Foe, f —oc and the lower semicontinuous proper convex functions f : R" —> R are nothing else than the W-convex functions, where W is the set of all affine functions on R".

73). 84) we obtain the so-called "duality inequality" a , 8. , see [236], thm. 69). 84) have been called by Rockafellar ([236], p. 19) "the central theorem about dual problems". , see [236], pp. 4-5). ,(0). , see [68], ch. 3, prop. 1). , see [236], thm. 16, or [68], ch. 3, prop. 2) that strong duality holds if and only if the subdifferential af (0) off at 0 is nonempty, and then af (0) coincides with the set of all optimal solutions WO of the dual problem (Q). Since the above classical approach to solving a primal problem (P) via a dual problem (Q) becomes ineffective in the case when a 0 p (at least, when the "duality gap" a — 18 > 0 is large), many modified Lagrangians have been introduced.

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 * ). 59) was introduced by Flachs and Pollatschek [88], as a tool for the study of duality for optimization problems involving minimum or maximum operations.