Showing papers in "Set-valued Analysis in 2006"
TL;DR: In this article, the authors investigated the regularity and stationarity properties of collections of sets in normed spaces, and provided a summary of different characterizations of regularity, and a list of sufficient conditions for a set to be regular.
Abstract: The paper continues investigations of stationarity and regularity properties of collections of sets in normed spaces It contains a summary of different characterizations (both primal and dual) of regularity and a list of sufficient conditions for a collection of sets to be regular
118 citations
TL;DR: In this paper, the differentiability properties of the projection onto the cone of positive semidefinite matrices was studied and the expression of the Clarke generalized Jacobian for the projection at any symmetric matrix is given.
Abstract: This paper studies the differentiability properties of the projection onto the cone of positive semidefinite matrices. In particular, the expression of the Clarke generalized Jacobian of the projection at any symmetric matrix is given.
42 citations
TL;DR: In this article, the authors investigated the properties of the proximal type algorithm and showed that the convergence rate of this algorithm is at least geometrical in the sense that the number of iterations is finite.
Abstract: Let H be a Hilbert space and A, B: H ⇉ H two maximal monotone operators. In this paper, we investigate the properties of the following proximal type algorithm:
$${{\left( {x_{{n + 2}} - 2x_{{n + 1}} + x_{n} } \right)}} \mathord{\left/ {\vphantom {{{\left( {x_{{n + 2}} - 2x_{{n + 1}} + x_{n} } \right)}} {\lambda ^{2}_{n} }}} \right. \kern-
ulldelimiterspace} {\lambda ^{2}_{n} } + A{\left( {{{\left( {x_{{n + 2}} - x_{{n + 1}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {x_{{n + 2}} - x_{{n + 1}} } \right)}} {\lambda _{n} }}} \right. \kern-
ulldelimiterspace} {\lambda _{n} }} \right)} + B{\left( {x_{{n + 2}} } \right)}
i 0,\,\,\,\,\,\,\,\,\,\,\,{\left( {\user1{\mathcal{A}}} \right)}$$
where (λ
n
) is a sequence of positive steps. Algorithm $${\left( {\user1{\mathcal{A}}} \right)}$$
may be viewed as the discretized equation of a nonlinear oscillator subject to friction. We prove that, if 0 ∈ int (A(0)) (condition of dry friction), then the sequence (x
n
) generated by $${\left( {\user1{\mathcal{A}}} \right)}$$
is strongly convergent and its limit x
∞ satisfies 0 ∈ A(0) + B(x
∞). We show that, under a general condition, the limit x
∞ is achieved in a finite number of iterations. When this condition is not satisfied, we prove in a rather large setting that the convergence rate is at least geometrical.
21 citations
TL;DR: Ge Geoffroy et al. as discussed by the authors investigated the stability of a cubic method with respect to some perturbations and showed that the pseudo-Lipschitzness of the map (f +- G)−1 is closely tied to the uniformity of the method.
Abstract: In Geoffroy et al, Acceleration of convergence in Dontchev's iterative method for solving variational inclusions Serdica Math. J.29 (2003), pp. 45–54] we showed the convergence of a cubic method for solving generalized equations of the form 0 ∈ f(x) +- G(x) where f is a\(\mathcal{C}^{{\text{2}}} \) function and G stands for a set-valued map. We investigate here the stability of such a method with respect to some perturbations. More precisely, we consider the perturbed equation y ∈ f(x) +- G(x) and we show that the pseudo-Lipschitzness of the map (f +- G)−1 is closely tied to the uniformity of our method in the sense that the attraction region does not depend on small perturbations of the parameter y. Finally, we provide an enhanced version of the convergence theorem established by Geoffroy, et al.
16 citations
TL;DR: The exponentiation theory of linear continuous operators on Banach spaces can be extended in manifold ways to a multivalued context in this paper, where the authors explore the Maclaurin exponentiation technique which is based on the use of a suitable power series.
Abstract: The exponentiation theory of linear continuous operators on Banach spaces can be extended in manifold ways to a multivalued context In this paper we explore the Maclaurin exponentiation technique which is based on the use of a suitable power series More precisely, we discuss about the existence and characterization of the Painleve–Kuratowski limit
$$[{\rm Exp}\;F](x)= \lim_{n\to\infty}\sum_{p=0}^n \frac{1}{p!}F^p(x)$$
under different assumptions on the multivalued map \(F\!:X\rightrightarrows X\) In Part II of this work we study the so-called recursive exponentiation method which uses as ingredient the set of trajectories associated to a discrete time evolution system governed by \(F\)
14 citations
TL;DR: In this article, the authors study sharp minima for multiobjective optimization problems and present sufficient and or necessary conditions for existence of such minima, some of which are new even in the single objective setting.
Abstract: We study sharp minima for multiobjective optimization problems. In terms of the Mordukhovich coderivative and the normal cone, we present sufficient and or necessary conditions for existence of such sharp minima, some of which are new even in the single objective setting.
14 citations
TL;DR: For maps φ on hyperspaces, the existence of semifixed sets, i.e., sets A satisfying one of the relations A ∩ φ(A), A ⊂ φ, A ∈ φ (A) ≠ ∅, is considered in this article.
Abstract: For maps φ on hyperspaces the existence of semifixed sets, i.e., of sets A satisfying one of the relations A ⊂ φ(A), A ⊃ φ(A), A ∩ φ(A) ≠ ∅, is considered. An application to set differential equations is also presented.
13 citations
TL;DR: The analysis and results go far beyond the particular context of convex processes considered by these authors, and are taking inspiration from a recent paper by Alvarez et al. on the relation between continuous and discrete time evolution systems.
Abstract: We continue with the exponentiation analysis of multivalued maps defined on Banach spaces. In Part I of this work we have explored the Maclaurin exponentiation technique which is based on the use of a suitable power series. Now we focus the attention on the so-called recursive exponentiation method. Recursive exponentials are specially useful when it comes to study the reachable set associated to a differential inclusion of the form \(\dot z \in F(z)\). The definition of the recursive exponential of \(F: X \rightrightarrows X\) uses as ingredient the set of trajectories associated to the discrete time system \(z_{k+1}\in F(z_k)\). Although we are taking inspiration from a recent paper by Alvarez et al. [1] on the relation between continuous and discrete time evolution systems, our analysis and results go far beyond the particular context of convex processes considered by these authors.
12 citations
TL;DR: In this paper, a new continuity concept called "supercalmness" is introduced, where arbitrarily small calmness constants can be obtained near the base point, which leads to essentially superlinear convergence results.
Abstract: Calmness of multifunctions is a well-studied concept of generalized continuity in which single-valued selections from the image sets of the multifunction exhibit a restricted type of local Lipschitz continuity where the base point is fixed as one point of comparison. Generalized continuity properties of multifunctions like calmness can be applied to convergence analysis when the multifunction appropriately represents the iterates generated by some algorithm. Since it involves an essentially linear relationship between input and output, calmness gives essentially linear convergence results when it is applied directly to convergence analysis. We introduce a new continuity concept called ‘supercalmness’ where arbitrarily small calmness constants can be obtained near the base point, which leads to essentially superlinear convergence results. We also explore partial supercalmness and use a well-known generalized derivative to characterize both when a multifunction is supercalm and when it is partially supercalm. To illustrate the value of such characterizations, we explore in detail a new example of a general primal sequential quadratic programming method for nonlinear programming and obtain verifiable conditions to ensure convergence at a superlinear rate.
11 citations
TL;DR: In this paper, the authors considered reachable sets of differential inclusions and gave sufficient conditions on the differential inclusion for the absolute continuity (of the integral) with respect to time and its weak derivative is formulated as a Hausdorff integral over the topological boundary.
Abstract: The well-known Reynold's Transport Theorem deals with the integral over a time-dependent set (that is evolving along a smooth vector field) and specifies its semiderivative with respect to time. Here reachable sets of differential inclusions are considered instead. Dispensing with any assumptions about the regularity of the compact initial set, we give sufficient conditions on the differential inclusion for the absolute continuity (of the integral) with respect to time and its weak derivative is formulated as a Hausdorff integral over the topological boundary.
9 citations
TL;DR: In this article, it was shown that the Painleve-Kuratowski limit for a special class of multivalued maps admits the representation [expF](x) = {eAx:A ∈ clco(Ξ)}. The operation of exponentiation has therefore a convexification effect on Ξ.
Abstract: This paper deals with the concept of exponentiability for a special class of multivalued maps. To be more precise, we discuss the exponentiability of a multivalued map F: X⇉X expressible in the form F(x) = {Ax:A ∈ Ξ}, with Ξ denoting a collection of linear continuous operators defined on a Banach space X. Among other results, we prove that, under suitable assumptions on Ξ, the Painleve–Kuratowski limit
$${\left[ {\exp F} \right]}{\left( x \right)} = {\mathop {\lim }\limits_{N \to \infty } }{\left( {I + \frac{1}{N}F} \right)}^{N} {\left( x \right)}$$
exists for all x ∈ X, and it admits the representation [expF](x) = {eAx:A ∈ clco(Ξ)}. The operation of exponentiation has therefore a convexification effect on Ξ. By exploiting the above-mentioned representation formula, we derive general properties for the semigroup {SF(t)}t⩾0 defined by
$$S_{F} {\left( t \right)}x = {\mathop {\lim }\limits_{N \to \infty } }{\left( {I + \frac{t}{N}F} \right)}^{N} {\left( x \right)}.$$
TL;DR: In this paper, the authors studied a nonconvex perturbed sweeping process with time delay in the infinite dimensional setting and proved that the problem has one and only one solution.
Abstract: This paper is devoted to the study of a nonconvex perturbed sweeping process with time delay in the infinite dimensional setting. On the one hand, the moving subset involved is assumed to be prox-regular and to move in an absolutely continuous way. On the other hand, the perturbation which contains the delay is single-valued, separately measurable, and separately Lipschitz. We prove, without any compactness assumption, that the problem has one and only one solution.
TL;DR: In this paper, it was shown that a composite Julia set generated by an infinite array of polynomial mappings is strongly analytic when regarded as a multifunction of the generating maps.
Abstract: It is shown that a composite Julia set generated by an infinite array of polynomial mappings is strongly analytic when regarded as a multifunction of the generating maps. An example of such a multifunction, the values of which have Holder Continuity Property, is constructed.
TL;DR: Theorem 3.1.1 as mentioned in this paper Theorem 2.1: Theorem 1.2: If one of the multifunctions X and Z is inner semicontinuous if and only if the other is, then one of them is inner semiconvariant.
Abstract: PROPOSITION 3.1. Assume the hypotheses of Theorem 2.1. Choose w0 2 W0 and take S 1⁄4 U0 fw0g. Define multifunctions X: U0 ! X0 and Z : U0 ! Z0 by requiring that ðu; xÞ belong to the graph of X if and only if ðu;w0; xÞ 2 MS, and that ðu; zÞ belong to the graph of Z if and only if ðu;w0; zÞ 2 NS, where MS and NS are defined by (7). Then one of the multifunctions X and Z is inner semicontinuous if and only if the other is. Proof. Let G be an open subset of X0 and suppose that Xðu0Þ meets G. Then there is x0 2 G such that ðu0;w0; x0Þ 2 MS. Let z0 1⁄4 ðu0;w0; x0Þ; then ðu0; w0; z0Þ 2 NS and therefore z0 2 Zðu0Þ; moreover, ðu0; z0Þ 1⁄4 x0. As is continuous, we can find an open neighborhood H of z0 and a neighborhood U 00 of u0 with U 00 U0 and such that if ðu; zÞ 2 U00 H then ðu; zÞ 2 G. As Z is inner semicontinuous, by shrinking U00 further if necessary we can ensure that if u 2 U00 then ZðuÞ meets H. Choose u 2 U00, and find z in ZðuÞ \ H. The triple ðu; w0; zÞ is then in NS, so the point ðu;w0; xÞ :1⁄4 ðu;w0; zÞ belongs to MS, and x 1⁄4 ðu; zÞ. As ðu; zÞ 2 U00 H we have x 1⁄4 ðu; zÞ 2 G; however, as ðu;w0; xÞ 2 MS we also have x 2 XðuÞ. Therefore XðuÞ meets G, so X is inner semicontinuous at u0. A parallel argument gives the proof in the other direction. I
TL;DR: In this paper, the authors established connections between the class of maximal monotone operators of Brondsted-rockafellar type and that of regular maximal monotonone operators, and showed that the relation between the two classes of operators can be inferred.
Abstract: The purpose of this paper is to establish connections between the class of maximal monotone operators of Brondsted–Rockafellar type and that of regular maximal monotone operators.
TL;DR: In this paper, the existence of solutions and necessary conditions of optimality for general minimization problems with constraints are studied and applications to an optimal control problem and a Lagrange multiplier rule are also given.
Abstract: The paper studies the existence of solutions and necessary conditions of optimality for a general minimization problem with constraints. Although we focus mainly on the case where the cost functional is locally Lipschitz, a general Palais–Smale condition is proposed and some of its properties are studied. Applications to an optimal control problem and a Lagrange multiplier rule are also given.
TL;DR: In this article, directional solutions to differential inclusions in Banach spaces were studied, which are based on the observation that for a differentiable function and a closed set, the probability of convergence is 0.
Abstract: We study a new type of solutions to differential inclusions in Banach spaces, which we call directional solutions. The idea is based on the observation that for a differentiable function \(u\) and a closed set \(V\)
$$u\prime {\left( t \right)} \in V\,{\text{iff}}\,{\mathop {\lim }\limits_{h \to 0} }d{\left( {\frac{{u{\left( {t + h} \right)} - u{\left( t \right)}}}{h},V} \right)} = 0.$$
TL;DR: In this paper, a subset A of a space X is C*-embedded in X if and only if for every continuous set-valued mapping φ of X into the non-empty compact subsets of a Banach space Y, every continuous selection g: A → Y for φ∣A is continuously extendable to the whole of X.
Abstract: We provide proper mapping-characterizations of some embedding-like properties weaker than \(P^{\lambda } \)-embedding. For instance, we show that a subset A of a space X is \(U^{\lambda } \)-embedded in X if and only if for every continuous map g: A → Y into a Banach space Y of weight w(Y) ⩽ λ, there exists a continuous set-valued mapping φ of X into the nonempty compact subsets of Y such that g is a selection for φ∣A (i.e., g(x) ∈ φ(x) for every x ∈ A). On the other hand, we show that a subset A is C*-embedded in X if and only if for every continuous set-valued mapping φ of X into the non-empty compact subsets of a Banach space Y, every continuous selection g: A → Y for φ∣A is continuously extendable to the whole of X. Combining both results we get the well-known mapping-characterization of \(P^{\lambda } \)-embedding which makes more transparent the relation ‘\(P^{\lambda } = U^{\lambda } + C^{*} \)’. Other weak components of \(P^{\lambda } \)-embedding are described in terms of expansions and selections, possible applications are demonstrated as well.
TL;DR: In this article, the authors studied class-qualified control problems where the weak derivatives of a weak derivative can be represented within a Baire function class and proved conditions under which the original and modified problems possess the same minimal values.
Abstract: We study multidimensional control problems involving first-order partial differential equations. To ensure the existence of sufficiently regular multipliers (from the space \({C^{\ast}}\)) in the first-order necessary optimality conditions, some restrictions of the feasible domain have to be added. In particular, we investigate ‘class-qualified’ problems where the weak derivatives of \(x\) can be represented within a Baire function class. In the present paper, we prove conditions under which the original and the modified problems possess the same minimal values.
TL;DR: In this article, the authors characterize cluster functions among arbitrary set-valued functions and show that every cluster function admits a selection h of Baire class 2 such that F = C(h;⋅).
Abstract: Given a single-valued function f between topological spaces X and Y, we interpret the cluster set C(f;x) as a multivalued function F=C(f;⋅) associated to f – the cluster function of f. For appropriate metrizable spaces X and Y, we characterize cluster functions C(f;⋅) among arbitrary set-valued functions F and show that every cluster function F=C(f;⋅) admits a selection h of Baire class 2 such that F=C(h;⋅).
TL;DR: A Banach space of dimension at least 2 does not admit an equi-Lipschitzian family of additive mappings parametrizing all non-empty compact convex sets as mentioned in this paper.
Abstract: A Banach space of dimension at least 2 does not admit an equi-Lipschitzian family of additive mappings parametrizing all non-empty compact convex sets. Examples of linear Lipschitzian as well as positively homogeneous equi-Lipschitzian parametrizations exist in the literature.