By Ivan Singer
This booklet examines summary convex research and offers the result of fresh examine, particularly on parametrizations of Minkowski style dualities and of conjugations of variety Lau. It explains the most techniques via situations and certain proofs.
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Additional info for Abstract Convex Analysis (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts)
In the present book we will not consider other existing concepts of generalized convexity of functions f : R" —> R that involve differentiability, such as pseudoconvexity, invexity, quasi-invexity, and pseudoinvexity, nor will we consider the generalizations of convex functions f : X —> R on a normed linear space X, such as paraconvex functions, since they are not defined in the abstract framework of functions f : X —> R on an arbitrary set X. 5 Dualities It is well-known that the classical concept of polarity from n-dimensional analytic geometry can be extended to subsets of two arbitrary sets X and W.
A4 -convex subsets of X; for example, as noted above, one can take M = B. 1b) to convexity systems B in a set X (see Chapter 2). Thus the theory of convexity systems and the intersectional (or the separational) approach to abstract convexity are equivalent. In the present book we will be more interested in the latter because of its applicability to optimization theory. 16) i=1 and using the following properties of (usual) convex hulls of any sets G and G: (1) if G C G, then co G C CO G; (2) G C CO G; (3) co (co G) = co G.
47)) and has assumed that ço (xo, w) = 0 for all w E W. 58) (with X* replaced by W) which is essentially equivalent to the general case of a pair (W, (p ), R is an arbitrary coupling function, where W is an arbitrary set and ço : X x W was given in . 91) to the primal problem (P). 85) remains valid also in the general case. 85). 100) wEW,dER). 89), we see that weak duality a = /3 is equivalent to the "stability" relation f (x 0 ) = fco (F)(x0). 7). 69)) is nonempty, and then f (x0) coincides with the set of all optimal solutions 14 of the dual problem fi, we have strong (Q).
Abstract Convex Analysis (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts) by Ivan Singer