Genericity and Porosity in Nonlinear Analysis and Optimization

We present several recent examples of the generic approach to nonlinear problems. In this approach, instead of considering, for instance, the convergence of an algorithm, one investigates an appropriate space of algorithms equipped with some natural complete metric and shows that convergence does indeed occur for most algorithms there. As we shall see, sometimes one can show that the complement of the good subset is not only of the first Baire category, but is also a sigma-porous set. 1 Convergence of Iterates Let (X, ρ) be a metric space. Recall that a subset E ⊂ X is called nowhere dense if the interior of its closure Ē is empty. Such a set can be considered “small”. A countable union of such sets, M = ⋃ {En : n = 1, 2, . . . }, is said to be of the first Baire category. The set M can also be considered “small”. All other sets are said to be of the second Baire category and are considered “large”. In 1899 René-Louis Baire proved that no complete metric space is of the first category. So in a complete metric space, the complement of a set of the first category is “large”. Let M = ⋃ {En : n = 1, 2, . . . }, where each En is nowhere dense, be of the first Baire category. Then its complement M ′ contains the set ⋂ {(Ēn) : n = 1, 2, . . . }, where each Gn = (Ēn) is open and (everywhere) dense. Thus we are led to consider the intersection of a sequence of dense, open subsets of X. Following Baire, we will see shortly that such an intersection is not only nonempty, but is, in fact, dense. In the proof we