Nonlinear error bounds for lower semicontinuous functions on metric spaces

As a development of the theory of linear error bounds for lower semicontinuous functions defined on complete metric spaces, introduced in Azé et al. (Nonlinear Anal 49, 643–670, 2002) and refined in Azé and Corvellec (ESAIM Control Optim Calc Var 10, 409–425, 2004), we propose a similar approach to nonlinear error bounds, based on the notion of strong slope, the variational principle, and the change-of-metric principle, the latter allowing to obtain sharp estimates for such error bounds through a reduction to the linear case.     

On error bounds for lower semicontinuous functions

We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents). For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f. In a reflexive Banach space X, we further obtain several sufficient and necessary conditions for the existence of error bounds in terms of the lower Dini derivative of f. Received: April 27, 2001 / Accepted: November 6, 2001?Published online April 12, 2002     

Linear and nonlinear error bounds for lower semicontinuous functions

In this paper, we give some characterizations of linear and nonlinear error bounds for lower semicontinuous functions by a new notion, called subslope. And, extend some results of Azé and Corvellec (SIAM J Optim 12:913–927, 2002) and Corvellec and Motreanu (Math Program Ser A 114:291–319, 2008) slightly. Furthermore, we get a sufficient and necessary condition for global linear error bounds.     

Epi-topology on spaces of lower semicontinuous functions into partially ordered spaces and hyperspaces

The concept of lower semicontinuity is extended to functions mapping into partially ordered spaces. A study is made of spaces of such lower semicontinuous functions under the epi-topology. These spaces are subspaces of hyperspaces with the Fell topology. The closure of such a function space in the hyperspace is characterized for certain spaces. A continuous selection theorem is established, showing that most such function spaces are not ech-complete.     

Error bounds for convex differentiable inequality systems in Banach spaces

The paper is devoted to studying the Hoffman global error bound for convex quadratic/affine inequality/equality systems in the context of Banach spaces. We prove that the global error bound holds if the Hoffman local error bound is satisfied for each subsystem at some point of the solution set of the system under consideration. This result is applied to establishing the equivalence between the Hoffman error bound and the Abadie qualification condition, as well as a general version of Wang &; Pang's result [30], on error bound of Hölderian type. The results in the present paper generalize and unify recent works by Luo &; Luo in [17], Li in [16] and Wang &; Pang in [30].     

A linear perturbed Palais-Smale condition for lower semicontinuous functions on Banach spaces

This paper introduces a notion of linear perturbed Palais-Smale condition for real-valued functions on Banach spaces. In terms of strongly exposed points, it presents a characterization which guarantees linear perturbed Palais-Smale condition holds for lower semicontinuous functions with bounded effective domains defined on a Banach space with the Radon-Nikody＇m property; and gives an example showing that linear perturbed P-S condition is strictly weaker than the P-S condition.     

Metric spaces of semicontinuous functions

This work was supported by the Russian Foundation of Fundamental Research under Grant No. 93-011-236.     

Taylor’s expansion for functions in Asplund spaces

The purpose of this paper is to present a version of second-order Taylor’s expansion for C^1,1 functions in Asplund spaces. An application is also given.     

Weak regularity of functions and sets in Asplund spaces

In this paper, we study a new concept of weak regularity of functions and sets in Asplund spaces. We show that this notion includes prox-regular functions, functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension. We establish also that weak regularity is equivalent to Mordukhovich regularity in finite dimension. Finally, we give characterizations of the weak regularity of epi-Lipschitzian sets in terms of their local representations.     

Generalized convexities of lower semicontinuous functions

In this paper a general theorem on the replacement of the condition “for all λ” in the definition of generalized convexity properties of lower semicontinuous functions by the condition “there exists a λ” is shown. This result can be applied to a number of special kinds of convexity and completes, for instance, studies of Behbikgeb concerning (explicitly) quasiconvex functions.     

Universal spaces of subdifferentials of sublinear operators ranging in the cone of bounded lower semicontinuous functions

We study Fréchet’s problem of the universal space for the subdifferentials ?P of continuous sublinear operators P: V → BC(X)_～ which are defined on separable Banach spaces V and range in the cone BC(X)_～ of bounded lower semicontinuous functions on a normal topological space X. We prove that the space of linear compact operators L ^c(? ², C(βX)) is universal in the topology of simple convergence. Here ? ² is a separable Hilbert space, and βX is the Stone-?ech compactification of X. We show that the images of subdifferentials are also subdifferentials of sublinear operators.     

A separation theorem for semicontinuous functions on preordered topological spaces

Suppose that X is a topological space with preorder , and that –g, f are bounded upper semicontinuous functions on X such that g(x) f(y) whenever x y. We consider the question whether there exists a bounded increasing continuous function h on X such that g h f, and obtain an existence theorem that gives necessary and sufficient conditions. This result leads to an extension theorem giving conditions that allow a bounded increasing continuous function defined on an open subset of X to be extended to a function of the same type on X. The application of these results to extremally disconnected locally compact spaces is studied.Received: 26 May 2004     

Asplund sets, differentiability and subdifferentiability of functions in Banach spaces

We show that Asplund sets are effective tools to study differentiability of Lipschitz functions, and ε-subdifferentiability of lower semicontinuous functions on general Banach spaces. If a locally Lipschitz function defined on an Asplund generated space has a minimal Clarke subdifferential mapping, then it is TBY-uniformly strictly differentiable on a dense Gδ subset of X. Examples are given of locally Lipschitz functions that are TBY-uniformly strictly differentiable everywhere, but nowhere Fréchet differentiable.     

Boundaries of Asplund spaces

We study the relationship between the classical combinatorial inequalities of Simons and the more recent (I)-property of Fonf and Lindenstrauss. We obtain a characterization of strong boundaries for Asplund spaces using the new concept of finitely self-predictable set. Strong properties for w^∗-K-analytic boundaries are established as well as a sup-lim sup theorem for Baire maps.     

Universality of Asplund spaces

Given any infinite cardinal , there exists no Banach space of density , which is Asplund or has the Point of Continuity Property and is universal for all reflexive spaces of density .

     

An open mapping theorem using lower semicontinuous functions

We give a simple and direct proof, using lower semicontinuous functions, of a generalization of the open mapping theorem for metrizable topological vector spaces (that are not necessarily locally convex) and operators with complete graph. Our result is in a form more applicable to applied (convex) analysis.     

Mazur intersection property for Asplund spaces

The main result of the present note states that it is consistent with the ZFC axioms of set theory (relying on Martin's Maximum MM axiom), that every Asplund space of density character ω₁ has a renorming with the Mazur intersection property. Combined with the previous result of Jiménez and Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain that the MIP renormability of Asplund spaces of density ω₁ is undecidable in ZFC.