Julian P. Revalski

Generic well-posedness of optimization problems in topological spaces

Let X be a completely regular Hausdorff topological space and let C ( X ) (the set of all real-valued bounded and continuous in X functions) be endowed with the sup-norm. Let s X , as usual, denotes the Stone-Cech compactification of X . We give a characterization of those X for which the set contains a dense -subset of C ( X ). These are just the spaces X which contain a dense Cech complete subspace. We call such spaces almost Cech complete. We also prove that X contains a dense completely metrizable subspace, if, and only if, C ( X ) contains a dense -subset of functions which determine Tykhonov well-posed optimization problems over X . For a compact Hausdorff topological space X the latter result was proved by Coban and Kenderov [CK1.CK2]. Relations between the well-posedness and Gâteaux and Frechet differentiability of convex functionals in C ( X ) are investigated. In particular it is shown that the sup-norm in C ( X ) is Frechet differentiable at the points of a dense -subset of C ( X ), if, and only if, the set of isolated points of X is dense in X . Conditions and examples are given when the set of points of Gateaux differentiability of the sup-norm in C ( X ) is a dense and Baire subspace of C ( X ) but does not contain a dense -subset of C ( X ).

Well-posedness by perturbations of variational problems

In this paper, we consider the extension of the notion of well-posedness by perturbations, introduced by Zolezzi for optimization problems, to other related variational problems like inclusion problems and fixed-point problems. Then, we study the conditions under which there is equivalence of the well-posedness in the above sense between different problems. Relations with the so-called diagonal well-posedness are also given. Finally, an application to staircase iteration methods is presented.

Porosity of ill-posed problems

We prove that in several classes of optimization problems, including the setting of smooth variational principles, the complement of the set of well-posed problems is σ-porous.

Enlargements and sums of monotone operators

In this paper we study two important notions related to monotone operators. One is the concept of enlargement of a given monotone operator which has turned out to be a useful tool in the analysis of approximate solutions to problems involving monotone operators. The second one is the notion of sum of monotone operators. For the latter we introduce and study a kind of extended sum of two monotone operators, which, in several important cases, turns out to be a maximal monotone operator.

Densely defined selections of multivalued mappings

Rather general suficient conditions are found for a given multivalued mapping F: X -* Y to possess an upper semicontinuous and compactvalued selection G which is defined on a dense G,-subset of the domain of F. The case when the selection G is single-valued (and continuous) is also investigated. The results are applied to prove some known as well as new results concerning generic differentiability of convex functions, Lavrentieff type theorem, generic well-posedness of optimization problems and generic nonmultivaluedness of metric projections and antiprojections.

A Variational Principle for Problems with Functional Constraints

In this paper we show that in several important classes of optimization problems, like mathematical programming with k-smooth data, quadratic programming in a Hilbert space, convex programming in a Banach space, semi-infinite programming, and optimal control of linear systems with quadratic cost, most of the problems (in the Baire category sense) are well-posed. This is derived from a general variational principle for problems with functional constraints.

Variational and Extended Sums of Monotone Operators

In this article we show that the notion of variational sum of maximal monotone operators, introduced by Attouch, Baillon and Thera in [3] in the setting of Hilbert spaces, can be successfully extended to the case of reflexive Banach spaces, preserving all of its properties. We make then a comparison with the usual pointwise sum and with the notion of extended sum proposed in our paper [26].

Generalized sums of monotone operators

Abstract This Note has two general aims: first to show that the notion of variational sum of maximal monotone operators, introduced by Attouch, Baillon and Thera in [1] in the setting of Hilbert spaces, can be extended to the case of reflexive Banach spaces, keeping all of its properties. And second, to compare it with the usual pointwise sum and with the notion of extended sum proposed in our paper [8].

Almost Every Convex or Quadratic Programming Problem Is Well Posed

We provide an abstract principle aimed at proving that classes of optimization problems aretypically well posed in the sense that the collection of ill-posed problems within each class is s-porous. As a consequence, we establish typical well-posedness in the above sense for unconstrained minimization of certain classes of functions (e.g., convex and quasi-convex continuous), as well as of convex programming with inequality constraints. We conclude the paper by showing that the collection of consistent ill-posed problems of quadratic programming of any fixed size has Lebesgue measure zero in the corresponding Euclidean space.

Generic Well-Posedness of Optimization Problems and the Banach-Mazur Game

Let X be a completely regular topological space. Denote, as usual, by C(X) the family of all bounded continuous real-valued functions in X. The space C(X) equipped with the sup-norm ||f||∞ = sup{| f(x)|: x ∈ X}, f ∈ C(X), becomes a Banach space. Each f ∈ C(X) determines a minimization problem: find x0 ∈ X with f(x 0) = inf {f(x) : x ∈ X} =: inf (X, f). We designate this problem by (X, f). Among the different properties of the minimization problem (X, f) the following ones are of special interest in the theory of optimization: (a) (X, f) has a solution (existence of the solution); (b) the solution set for (X, f) is a singleton (uniqueness of the solution); (c) if f(x*) is close to inf (X, f), then x* is a good approximation of the solution of (X, f) (stability of the solution—see bellow the precise definition).