arXiv subjects

Enrico Miglierina

Publications and source records attributed to Enrico Miglierina.

18 recordsLinked to original sources

arXiv2026

Characterization of regularity via variational stability of alternating projections sequences

The notion of regular pair $(A,B)$ for two nonempty closed convex subsets $A$ and~$B$ of a Hilbert space $\H$ was introduced by Borwein and Bauschke in 1993 to ensure convergence (in norm) of the alternating projection method to some point of the best approximation set. In 2022, De Bernardi and Miglierina showed that regularity of the pair $(A,B)$ guarantees, additionally, the convergence for any variational perturbation of the alternating projection method, provided the corresponding best approximation sets are bounded. In this work, we show that the converse assertion is also true. Moreover, this converse assertion holds without requiring the best approximation sets to be bounded.

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arXiv2024

A note on the Moment Problem for codimension greater than 1

We provide new conditions under which the alternating projection sequence converges in norm for the convex feasibility problem where a linear subspace with finite codimension $N\geq 2$ and a lattice cone in a Hilbert space are considered. The first result holds for any Hilbert lattice, assuming that the orthogonal of the linear subspace admits a basis made by disjoint vectors with respect to the lattice structure. The second result is specific for $\ell^2(\mathbb{N})$ and is proved when only one vector of the basis is not in the cone but the sign of its components is definitively constant and its support has finite intersection with the supports of the remaining vectors.

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arXiv2024

ABB theorems: Results and limitations in infinite dimensions

We construct a weakly compact convex subset of $\ell^2$ with nonempty interior that has an isolated maximal element, with respect to the lattice order $\ell _+^2$. Moreover, the maximal point cannot be supported by any strictly positive functional, showing that the Arrow-Barankin-Blackwell theorem fails. This example discloses the pertinence of the assumption that the cone has a bounded base for the validity of the result in infinite dimensions. Under this latter assumption, the equivalence of the notions of strict maximality and maximality is established

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arXiv2024

Subspaces of separable $L_1$-preduals: $W_α$ everywhere

The spaces $W_α$ are the Banach spaces whose duals are isometric to $\ell_1$ and such that the standard basis of $\ell_1$ is $w^*$-convergent to $α\in \ell_1$. The core result of our paper proves that an $\ell_1$-predual $X$ contains isometric copies of all $W_α$, where the norm of $α$ is controlled by the supremum of the norms of the $w^*$-cluster points of the extreme points of the closed unit ball in $\ell_1$. More precisely, for every $\ell_1$-predual $X$ we have $$ r^*(X)=\sup\left\lbrace \left\|g^*\right\|: g^*\in \left(\mathrm{ext}\, B_{\ell_1}\right)'\right\rbrace =\sup \left\lbrace \left\| α\right\|: \, α\in B_{\ell_1}, \, W_α\subset X\right\rbrace . $$ We also prove that, for any $\varepsilon >0$, $X$ contains an isometric copy of some space $W_α$ with $\left\| α\right\|>r^*(X)- \varepsilon$ which is $(1+ \varepsilon)$-complemented in $X$. From these results we obtain several consequences. First we provide a new characterization of separable $L_1$-preduals containing an isometric copy of a space of affine continuous functions on a Choquet simplex. Then, we prove that an $\ell_1$-predual $X$ contains almost isometric copies of the space $c$ of convergent sequences if and only if $X^*$ lacks the stable $w^*$-fixed point property for nonexpansive mappings.

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arXiv2023

On Projected Solutions for Quasi Equilibrium Problems with Non-self Constraint Map

In a normed space setting, this paper studies the conditions under which the projected solutions to a quasi equilibrium problem with non-self constraint map exist. Our approach is based on an iterative algorithm which gives rise to a sequence such that, under the assumption of asymptotic regularity, its limit points are projected solutions. Finally, as a particular case, we discuss the existence of projected solutions to a quasi variational inequality problem.

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arXiv2023

Stochastic Approximation in convex multiobjective optimization

Given a strictly convex multiobjective optimization problem with objective functions $f_1,\dots,f_N$, let us denote by $x_0$ its solution, obtained as minimum point of the linear scalarized problem, where the objective function is the convex combination of $f_1,\dots,f_N$ with weights $t_1,\ldots,t_N$. The main result of this paper gives an estimation of the averaged error that we make if we approximate $x_0$ with the minimum point of the convex combinations of $n$ functions, chosen among $f_1,\dots,f_N$, with probabilities $t_1,\ldots,t_N$, respectively, and weighted with the same coefficient $1/n$. In particular, we prove that the averaged error considered above converges to 0 as $n$ goes to $\infty$, uniformly w.r.t. the weights $t_1,\ldots,t_N$. The key tool in the proof of our stochastic approximation theorem is a geometrical property, called by us small diameter property, ensuring that the minimum point of a convex combination of the function $f_1,\dots,f_N$ continuously depends on the coefficients of the convex combination.

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arXiv2022

Explicit models of $\ell_1$-preduals and the weak$^*$ fixed point property in $\ell_1$

We provide a concrete isometric description of all the preduals of $\ell_1$ for which the standard basis in $\ell_1$ has a finite number of $w^*$-limit points. Then, we apply this result to give an example of an $\ell_1$-predual $X$ such that its dual $X^*$ lacks the weak$^*$ fixed point property for nonexpansive mappings (briefly, $w^*$-FPP), but $X$ does not contain an isometric copy of any hyperplane $W_α$ of the space $c$ of convergent sequences such that $W_α$ is a predual of $\ell_1$ and $W_α^*$ lacks the $w^*$-FPP. This answers a question left open in the 2017 paper of the present authors.

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arXiv2020

Regularity and stability for a convex feasibility problem

Let us consider two sequences of closed convex sets $\{A_n\}$ and $\{B_n\}$ converging with respect to the Attouch-Wets convergence to $A$ and $B$, respectively. Given a starting point $a_0$, we consider the sequences of points obtained by projecting on the "perturbed" sets, i.e., the sequences $\{a_n\}$ and $\{b_n\}$ defined inductively by $b_n=P_{B_n}(a_{n-1})$ and $a_n=P_{A_n}(b_n)$. Suppose that $A\cap B$ (or a suitable substitute if $A \cap B=\emptyset$) is bounded, we prove that if the couple $(A,B)$ is (boundedly) regular then the couple $(A,B)$ is $d$-stable, i.e., for each $\{a_n\}$ and $\{b_n\}$ as above we have $\mathrm{dist}(a_n,A\cap B)\to 0$ and $\mathrm{dist}(b_n,A\cap B)\to 0$.

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arXiv2020

A variational approach to the alternating projections method

The 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex sets $A$ and $B$ in a Hilbert space $X$. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of sets, each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to $A$ and $B$. Given a starting point $a_0$, we consider the sequences of points obtained by projecting on the "perturbed" sets, i.e., the sequences $\{a_n\}$ and $\{b_n\}$ given by $b_n=P_{B_n}(a_{n-1})$ and $a_n=P_{A_n}(b_n)$. Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences $\{a_n\}$ and $\{b_n\}$ converge in norm to a point in the intersection of $A$ and $B$. In particular, we consider both when the intersection $A\cap B$ reduces to a singleton and when the interior of $A \cap B$ is nonempty. Finally we consider the case in which the limit sets $A$ and $B$ are subspaces.

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arXiv2019

Weak$^*$ fixed point property and the space of affine functions

First we prove that if a separable Banach space $X$ contains an isometric copy of an infinite-dimensional space $A(S)$ of affine continuous functions on a Choquet simplex $S$, then its dual $X^*$ lacks the weak$^*$ fixed point property for nonexpansive mappings. Then, we show that the dual of a separable Lindenstrauss space $X$ fails the weak$^*$ fixed point property for nonexpansive mappings if and only if $X$ has a quotient isometric to some space $A(S)$. Moreover, we provide an example showing that "quotient" cannot be replaced by "subspace". Finally, it is worth to be mentioned that in our characterization the space $A(S)$ cannot be substituted by any space $\mathcal{C}(K)$ of continuous functions on a compact Hausdorff $K$.

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arXiv2018

Stability of a convex feasibility problem

The 2-sets convex feasibility problem aims at finding a point in the intersection of two closed convex sets $A$ and $B$ in a normed space $X$. More generally, we can consider the problem of finding (if possible) two points in $A$ and $B$, respectively, which minimize the distance between the sets. In the present paper, we study some stability properties for the convex feasibility problem: we consider two sequences of sets, each of them converging, with respect to a suitable notion of set convergence, respectively, to $A$ and $B$. Under appropriate assumptions on the original problem, we ensure that the solutions of the perturbed problems converge to a solution of the original problem. We consider both the finite-dimensional and the infinite-dimensional case. Moreover, we provide several examples that point out the role of our assumptions in the obtained results.

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arXiv2016

Stability constants of the weak$^*$ fixed point property for the space $\ell_1$

The main aim of the paper is to study some quantitative aspects of the stability of the weak$^*$ fixed point property for nonexpansive maps in $\ell_1$ (shortly, $w^*$-fpp). We focus on two complementary approaches to this topic. First, given a predual $X$ of $\ell_1$ such that the $σ(\ell_1,X)$-fpp holds, we precisely establish how far, with respect to the Banach-Mazur distance, we can move from $X$ without losing the $w^*$-fpp. The interesting point to note here is that our estimate depends only on the smallest radius of the ball in $\ell_1$ containing all $σ(\ell_1,X)$-cluster points of the extreme points of the unit ball. Second, we pass to consider the stability of the $w^*$-fpp in the restricted framework of preduals of $\ell_1$. Namely, we show that every predual $X$ of $\ell_1$ with a distance from $c_0$ strictly less than $3$, induces a weak$^*$ topology on $\ell_1$ such that the $σ(\ell_1,X)$-fpp holds.

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arXiv2016

Weak$^*$ Fixed Point Property in $\ell_1$ and Polyhedrality in Lindenstrauss Spaces

The aim of this paper is to study the $w^*$-fixed point property for nonexpansive mappings in the duals of separable Lindenstrauss spaces by means of suitable geometrical properties of the dual ball. First we show that a property concerning the behaviour of a class of $w^*$-closed subsets of the dual sphere is equivalent to the $w^*$-fixed point property. Then, the main result of our paper shows an equivalence between another, stronger geometrical property of the dual ball and the stable $w^*$-fixed point property. The last geometrical notion was introduced by Fonf and Veselý as a strengthening of the notion of polyhedrality. In the last section we show that also the first geometrical assumption that we have introduced can be related to a polyhedral concept for the predual space. Indeed, we give a hierarchical structure among various polyhedrality notions in the framework of Lindenstrauss spaces. Finally, as a by-product, we obtain an improvement of an old result about the norm-preserving compact extension of compact operators.

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arXiv2015

Rethinking Polyhedrality for Lindenstrauss Spaces

A recent example by the authors (see arXiv:1503.09088 [math.FA]) shows that an old result of Zippin about the existence of an isometric copy of $c$ in a separable Lindenstrauss space is incorrect. The same example proves that some characterizations of polyhedral Lindenstrauss spaces, based on the result of Zippin, are false. The main result of the present paper provides a new characterization of polyhedrality for the preduals of $\ell_{1}$ and gives a correct proof for one of the older. Indeed, we prove that for a space $X$ such that $X^{*}=\ell_{1}$ the following properties are equivalent: (1) $X$ is a polyhedral space; (2) $X$ does not contain an isometric copy of $c$; (3) $\sup\left\{ x^{*}(x)\,:\, x^{*}\in\mathrm{ext}\left(B_{X^{*}}\right)\setminus D(x)\right\} <1$ for each $x\in S_{X}$, where $D(x)=\left\{ x^{*}\in S_{X^{*}}:x^{*}(x)=1\right\}$. By known theory, from our result follows that a generic Lindenstrauss space is polyhedral if and only if it does not contain an isometric copy of $c$. Moreover, a correct version of the result of Zippin is derived as a corollary of the main result.

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arXiv2015

A remark on spaces of affine continuous functions on a simplex

We present an example of an infinite dimensional separable space of affine continuous functions on a Choquet simplex that does not contain a subspace linearly isometric to $c$. This example disproves a result stated in M. Zippin. On some subspaces of Banach spaces whose duals are $L_1$ spaces. Proc. Amer. Math. Soc. 23, (1969), 378-385.

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arXiv2015

Separable Lindenstrauss spaces whose duals lack the weak$^*$ fixed point property for nonexpansive mappings

In this paper we study the $w^*$-fixed point property for nonexpansive mappings. First we show that the dual space $X^*$ lacks the $w^*$-fixed point property whenever $X$ contains an isometric copy of the space $c$. Then, the main result of our paper provides several characterizations of weak-star topologies that fail the fixed point property for nonexpansive mappings in $\ell_1$ space. This result allows us to obtain a characterization of all separable Lindenstrauss spaces $X$ inducing the failure of $w^*$-fixed point property in $X^*$.

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arXiv2012

Reflexive Cones

Reflexive cones in Banach spaces are cones with weakly compact intersection with the unit ball. In this paper we study the structure of this class of cones. We investigate the relations between the notion of reflexive cones and the properties of their bases. This allows us to prove a characterization of reflexive cones in term of the absence of a subcone isomorphic to the positive cone of \ell_{1}. Moreover, the properties of some specific classes of reflexive cones are investigated. Namely, we consider the reflexive cones such that the intersection with the unit ball is norm compact, those generated by a Schauder basis and the reflexive cones regarded as ordering cones in a Banach spaces. Finally, it is worth to point out that a characterization of reflexive spaces and also of the Schur spaces by the properties of reflexive cones is given.

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arXiv2012

Some results on condition numbers in convex multiobjective optimization

Various notions of condition numbers are used to study some sensitivity aspects of scalar optimization problems. The aim of this paper is to introduce a notion of condition number to study the case of a multiobjective optimization problem defined via m convex C^1,1 objective functions on a given closed ball in R^n. Two approaches are proposed: the first one adopts a local point of view around a given solution point, whereas the second one considers the solution set as a whole. A comparison between the two notions of well-conditioned problem is developed. We underline that both the condition numbers introduced in the present work reduce to the same of condition number proposed by Zolezzi in 2003, in the special case of the scalar optimization problem considered there. A pseudodistance between functions is defined such that the condition number provides an upper bound on how far from a well-conditioned function f a perturbed function g can be chosen in order that g is well-conditioned too. For both the local and the global approach an extension of classical Eckart-Young distance theorem is proved, even if only a special class of perturbations is considered.

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