R is any coupling function, replacing w(x) by ga(x, w). 13).
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). e. 65), is equivalent to wo E aço f (x0). 90) (with arbitrary (W, ço)). In turn the general theory obtained in this way can be applied not only to convex optimization but also to a large number of other (known and new) cases. 92) encompasses, as particular cases, various modified Lagrangians.
The theory of quasi-convexity of functions on R" can be extended to the abstract framework of functions on a set X. 40) be "convex" subsets of X, in the given sense. We will study these concepts of abstract quasi-convexity of functions on a set X in Chapter 4. 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.
Abstract Convex Analysis (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts) by Ivan Singer