In the present paper, the problem of estimating the contingent cone to the solution set associated with certain set-valued inclusions is addressed by variational analysis methods and tools. As a main result, inner (resp. outer) approximations, which are expressed in terms of outer (resp. inner) prederivatives of the set-valued term appearing in the inclusion problem, are provided. For the analysis of inner approximations, the evidence arises that the metric increase property for set-valued mappings turns out to play a crucial role. Some of the results obtained in this context are then exploited for formulating necessary optimality conditions for constrained problems, whose feasible region is defined by a set-valued inclusion.
1 Introduction and problem statement
The present work aims at providing elements for a first-order analysis of the solutions to set-valued inclusions. By set-valued inclusion the following problem is meant: given a set-valued mapping , a nonempty closed set , and a closed, convex and pointed cone , with ,
The solution set to (SVI) will be denoted throughout the paper by
Some motivations for considering such problems as (SVI), mainly coming from the robust approach to uncertain optimization as well as from mathematical economics, are discussed in [25, 26]. It is clear that can be nominally expressed in terms of upper inverse image of C through F, i.e. , resulting in . In spite of such a simple reformulation, may happen to be a rather involved set, as reflecting the complicated nature that set-valued mappings may often exhibit. Therefore, in order to glean information on the local geometry of , it becomes crucial to undertake a systematic study of first-order approximations of it. In the present paper, such a task will be pursued by focusing on the contingent (a.k.a. Bouligand–Severi tangent) cone to . Among many local conic approximations of sets currently at disposal in set-valued analysis, this is one of the mostly employed and widely investigated. It plays an essential role in constructing derivatives for set-valued mappings through a graphical approach (see, for instance, [2, 23]) and it emerges as a basic tool in formulating first-order optimality conditions (see, among others, [11, 18, 23]).
The line of thought behind the analysis here proposed is that a workable representation of the contingent cone to could be obtained by the upper inverse image of C through a first-order approximation of F. In other words, this amounts to considering the interchange of two operations of different nature: namely, on one hand the approximation of sets and mappings and, on the other hand, the operation of taking the upper inverse. It is worth noting that the approach stemming from this line of thought shares the spirit of the celebrated Lyusternik theorem about the representation of the tangent space to a smooth manifold, which is expressed by an equation system (see ). In its modern formulation, suitable for problems of the form
where is a given single-valued mapping, this theorem states that, under proper regularity assumptions valid at a solution , the representation
holds, where stands for the strict derivative of f at (see [23, Theorem 11.4.4]). In a similar manner, the present investigations explore the possibility of exploiting the upper inverse image of first-order approximations of F for providing representations of . If acts as a first-order approximation of F near , one expects that geometric properties of H result in an easy geometry of the set . For example, it is well known that if H is positively homogeneous, then is a cone; if H is concave (in the sense of [26, Definition 2.3]), then is a convex set. Such correspondences evidently contribute to a better understanding of the local structure of .
In developing the above proposed approach of analysis, passing from such problems as (1.1) to (SVI), a methodological question to face is which approximation tool is to be used for the term F. Since the fact that is a solution to (SVI) involves all elements in , such an approximation tool should not be based on the local behavior of F around a reference element of its graph (as it happens with graphical derivatives and coderivatives [2, 12, 17]), but should take into account the whole set . For this reason, the present approach utilizes the notion of prederivative (see [13, 14, 20]). The splitting of this notion into an outer and an inner version allows one to study separately the question of inner and outer tangential approximation of . Prederivatives are not the only set-oriented derivative-like notion for set-valued mappings, that is able to take into account the whole image of F at a reference point. A different construction, which relies on the Rådström embedding theorem, can be found in . An intrinsic limitation of the notion of π-differentiability there proposed consists in referring to mappings with convex and bounded values. Since in the present analysis the set-valued mapping F considered in (SVI) will not be required to satisfy that assumption, a line of research exploiting π-differentiability is left open for future investigations, which will focus on more particular classes of (SVI).
To the best of the author’s knowledge, the study of the solution analysis of (SVI) was initiated in . Some advances in this direction, already including representations of the contingent cone to , have been recently obtained in , under assumptions of concavity of F and boundedness of its values. A feature distinguishing the present investigations is the essential employment of the metric C-increase property for set-valued mappings, while avoiding any concavity assumption on F. This property describes a certain behavior of mappings that links the metric structure of the domain with the cone C appearing in (SVI). Roughly speaking, it can be viewed as a counterpart, valid in partially ordered vector spaces, of the so-called decrease principle for scalar functions, in use in variational analysis (see [7, Chapter 3.6] and [21, Chapter 1.6]). It is well known that for traditional equation systems and, to a certain extent, for generalized equations of the form (1.1), open covering (and hence, metric regularity) is the main property for mappings ensuring local solvability and, as such, it became the key concept to achieve tangential approximations of solution sets. In a similar manner, the metric C-increase property turns out to be a key concept in order to establish a proper error bound for (SVI) and, through such kind of estimate, to get the inner tangential approximation of .
The contents of the paper are arranged in the subsequent sections as follows. In Section 2, the major analysis tools needed to develop the approach analysis summarized above are presented with references. Essentially, all of them are well-known notions and facts from set-valued analysis and generalized differentiation, with the only exception of the metric C-increase property, to which a specific subsection is devoted. In Section 3, the main contributions of the paper, which concern inner and outer tangential approximations of , are established and discussed. In Section 4, optimization problems, whose feasible region is defined by set-valued inclusions (SVI), are considered and some of the results achieved in Section 3 are exploited for deriving necessary optimality conditions suitable for problems of that form. Such an application may serve as an evidence of the fact that “the calculus of tangents is one of the main techniques of optimization” (as stated in ).
2 Analysis tools
The notation in use throughout the paper is standard: and denote the natural and the real number set, respectively, denotes the nonnegative orthant in the Euclidean space , whose (Euclidean) norm is indicated by . The null vector in any Euclidean space is indicated by . Given an element x of a metric space and a nonnegative real r, denotes the closed ball with center x and radius r. In particular, if , then and stand for the unit ball and the unit sphere, respectively. Given a subset S of an Euclidean space, by and the topological interior and the boundary of S are denoted, respectively, whereas denotes the convex hull of S. By the distance of x from a subset is denoted, with the convention that . The r-enlargement of a set is indicated by . Given a pair of subsets , the symbol denotes the excess of over , where the convention is accepted. The symbol indicates the Pompeiu–Hausdorff distance between and .
Whenever is a set-valued mapping,
denote the graph and the domain of F, respectively. All set-valued mappings appearing in the paper will be supposed to take closed values, unless otherwise stated. This fact will be implicitly assumed, in particular, with reference to mappings resulting from the sum of set-valued mappings. Moreover, indicates the space of all linear mappings acting from to , endowed with the operator norm . If , then denotes the adjoint operator to Λ. Given a function , where X is a given set, and stand for the 0-sublevel and the strict 0-superlevel set of φ, respectively. Other notations will be explained contextually to their use.
Throughout the text, the acronyms l.s.c. and p.h. stand for lower semicontinuous and positively homogeneous, respectively.
2.1 Elements of set-valued and variational analysis
Let us recall that a set-valued mapping is said to be l.s.c. at if for every open set such that there exists such that
The mapping is said to be Lipschitz (continuous) with constant if
A known fact which is relevant to the present analysis is that if a set-valued mapping is l.s.c. at each point of , then is a closed set for every closed set C (see, for instance, [1, Lemma 17.5]). This fact makes it clear that, under the assumptions made on the problem data of (SVI) (namely, S closed and C closed, convex and pointed cone), if F is a l.s.c. set-valued mapping, then the solution set is a closed subset (possibly empty) of .
In studying the variational behavior of set-valued mappings, a basic tool of analysis is the excess of a set over another. The following remark gathers several known facts concerning the behavior of the excess, which will be employed in the subsequent analysis (for their proof, whenever not trivial, one can refer to ).
Let be nonempty and let be a closed, convex cone.
If , for any it holds (additive behavior of the excess with respect to enlargements).
If , it holds (invariance of the excess under conic extension).
Let . It holds .
(i) Given a set-valued mapping , it is known that the aforementioned semicontinuity property of F implies a corresponding semicontinuity property of the excess function , which is associated with F and C, namely
In other words, if F is l.s.c. at , then ϕ is l.s.c. at (the proof can be found in [25, Lemma 2.3]).
(ii) Since C is a closed set, it is clear that if and only if . Therefore, the solution set to (SVI) can be characterized, via the function ϕ, in the following terms: .
The main result of this paper will be achieved by means of an error bound estimate for the solution set to (SVI). The technique of proof of the latter one relies on the characterization of error bounds for l.s.c. functions on a complete metric space through the notion of strong slope (see, among others, [4, 3]). Let us recall that, after , given a function defined on a metric space and , the strong slope of φ at is defined as the quantity
Notice that, if as a metric space X one takes a closed subset containing , the above definition becomes
For the purposes of the present work, the following general condition for an error bound, which can be obtained as a special case of [3, Corollary 3.1], will be employed.
Let be a complete metric space, let be a function l.s.c. on X, and let . Suppose that and are such that
Then it holds
2.2 The metric C-increase property
The next definition introduces the main property of set-valued mappings, on which the proposed approach to the solution analysis of (SVI) relies. It postulates a behavior of mappings that links the metric structure of the domain with the partial ordering induced on the range space by the cone C in the standard way (henceforth denoted by ), i.e. if and only if .
Definition 2.4 (Metrically C-increasing mapping).
Let be a nonempty closed set and let be a closed, convex cone, with . Consider a set-valued mapping :
F is said to be metrically C-increasing around relative to S if there exist and such that(2.2)
is called exact bound of metric C-increase of F around , relative to S.
F is said to be globally metrically C-increasing if there exists such that(2.3)
is called global exact bound of metric C-increase of F.
As a comment to the above property, let us observe that the behavior that it describes can be regarded as a set-valued version of a phenomenon, which in the case of scalar functions is known as decrease principle of variational analysis. By this term, any condition is denoted which ensures the existence of a constant such that
where is a l.s.c. and bounded from below function defined on a proper (at least, metric) space, is a reference point and . Often, such a condition finds a formulation in terms of Fréchet subdifferential, provided that X is a Fréchet smooth Banach space (see [7, Theorem 3.6.2]), or, more generally, in terms of strong slope, if X is a complete metric space (see ). The decrease principle appeared as a fundamental tool in the analysis of error bounds and solution stability for inequalities and, as such, it plays a key role in establishing implicit multifunction theorems (see ). This led the author to employ the term “metric C-increase” in .
(i) Whenever , the notion of metric C-increase around , relative to S, reduces to the notion of local metric C-increase around , as defined in .
(ii) An equivalent reformulation of the inclusion (2.2) that will be useful is clearly
Let be defined by
and let . By a direct check of Definition 2.4 (ii), one can see that the set-valued mapping F is globally metrically -increasing, with .
Other examples of classes of metrically C-increasing set-valued mappings, along with verifiable conditions for detecting such property, will be provided in the next subsection. Further examples can be found in .
Below, the aforementioned error bound condition, instrumental to the solution analysis of (SVI), is established. Such a condition can be viewed as a refinement of [25, Theorem 4.3]. It is presented here with its full proof, because it turns out that, by using a more adequate technique of proof, one assumption made in the mentioned theorem can be dropped out and the whole argument gains in clearness.
Lemma 2.7 (Local error bound under metric C-increase).
Suppose that is a set-valued mapping, is a closed convex cone, and S is a closed set defining a problem (SVI), with . Suppose the following conditions:
F is l.s.c. in for some .
F is metrically C -increasing around , relatively to S.
Then, for every , there exists such that
Consider the function , defined as in (2.1). According to Remark 2.2 (i), ϕ is l.s.c. on by virtue of hypothesis (i). Take . Without any loss of generality, it is possible to assume that δ is smaller than the value of δ appearing in inclusion (2.2) of Definition 2.4. Now, fix an arbitrary . Then, according to hypothesis (ii), taken any such that
there exists such that
The last inequality chain implies
so x can not be a local minimizer of ϕ over S. By consequence, when calculating the strong slope of ϕ at x in the metric space S, one finds
This shows that
Since S, as a closed subset of , is a complete metric space, Proposition 2.3 guarantees that
Thus, it suffices to set to achieve the thesis. ∎
The following example illustrates the essential role played by the metric C-increase property for the validity of the error bound (2.4).
Example 2.8 (Error bound failure).
Consider the set-valued mapping defined by
and take , and . With these data, the resulting (SVI) evidently admits as a solution set. Therefore, one has
On the other hand, one sees that it is
As a consequence, for any , the error bound inequality
fails to hold in any interval , whatever the value of is. Observe that F is l.s.c. in a neighborhood of 0, so hypothesis (i) of Lemma 2.7 is fulfilled. Instead, F is not metrically -increasing around 0, relative to (in other terms, locally metrically -increasing around 0).
2.3 Generalized differentiation tools
Let be a nonempty closed set and let . As a first-order approximation of S near , the following different cones will be used:
They are called the contingent cone, the feasible direction cone, the weak feasible direction cone and the Clarke tangent cone to S at , respectively (see, for instance, [2, 11, 23]). It is to be noted that the above definition of Clarke tangent cone is actually an equivalent reformulation provided in [11, Proposition 2.2] of the original notion. The following relations of inclusion are known to hold in general:
When, in particular, S is locally convex around , i.e. there exists such that is convex, then
(see, for instance, [23, Proposition 11.1.2 (d)]). The Clarke tangent cone is always closed and convex (see [11, Proposition 2.3]). The contingent cone, introduced in [8, 24], will be the main object of study in the present analysis. It follows from its very definition that it is determined only by the geometric shape of a set near the reference point, namely for any it is
Of course, whenever S is a closed convex cone, one finds .
Given a nonempty and , the following characterization of in terms of the Dini lower derivative of the function at will be useful:
Given a cone , let us recall that the set
is called (negative) dual cone of C. Whenever S is locally convex around (and hence is convex), such an operator is connected with the normal cone to S at in the sense of convex analysis by the following well-known relation:
Let be a function which is finite around . Following , the sets
are called the Fréchet subdifferential of φ at and the Fréchet upper subdifferential of φ at , respectively. It is readily seen that, whenever φ is (Fréchet) differentiable at , then , whereas whenever is convex (resp. concave), the set (resp. ) reduces to the subdifferential (resp. superdifferential) of φ at in the sense of convex analysis.
The following variational description of the Fréchet upper subdifferential of φ at will be exploited in the sequel: for every there exists a function , differentiable at and with , such that for every and (see [17, Theorem 1.88]).
While cones are the basic objects for approximating sets, positively homogeneous set-valued mappings are the basic tools for approximating multifunctions. Recall that a set-valued mapping is positively homogeneous (for short, p.h.) if and
Within the class of p.h. set-valued mappings, fans will play a prominent role in the present analysis (see ).
Definition 2.11 (Fan).
A set-valued mapping is said to be a fan if it fulfils the following conditions:
It is p.h.
It is convex-valued.
Fans are set-valued mappings with a useful geometric structure, arising in a large variety of contexts. It is clear that the class of all fans acting between and includes, as a very special case, the space . An important class of fan, playing a role in the present analysis, is discussed below.
Example 2.12 (Fans generated by linear mappings).
Let be a nonempty, convex and closed set. The set-valued mapping defined by
is known to be a fan (see ). In such a circumstance, the set will be called a generator for H. In particular, whenever is a polytope in , the fan generated by will be said to be finitely-generated. For example, in the case , one may take the class of all linear mappings represented by doubly stochastic matrices. After the Birkhoff–von Neumann theorem, this class is known to be a polytope, resulting from the convex hull of all the permutation matrices, which are its extreme elements (see ). Note that any finitely-generated fan takes compact values which are polytopes in the range space . In general, for any fan H generated by linear mappings it must be .
The set-valued mapping , defined by
is a fan. Since holds, it is clear that H can not be generated by any set .
Further examples of fans are provided in .
According to the present approach of analysis, the upper inverse image of C through a given fan will be a key element to express the tangential approximation of . In this perspective, the next remark gathers some elementary algebraic/topological properties of such a set.
(i) It is plain to see that if is a fan and is a closed convex cone, then the set is a convex cone (possibly empty). Notice that, in general, may happen to be not closed. For example, if taking and such a fan as defined in Example 2.13, one finds (consistently, H fails to be l.s.c. at 0).
(ii) It is worth noting that, in the case of a fan generated by a set , it results in
As an immediate consequence of the last equality, one deduces that the convex cone is closed whenever H is a fan generated by linear mappings. Furthermore, if a fan H is finitely-generated, i.e.
with , for , then it results in
In this case, each set turns out to be polyhedral, provided that C is so, and therefore inherits a polyhedral cone structure.
(iii) Whenever is a fan generated by a bounded set , it turns out to be Lipschitz. More precisely, if , it holds
Indeed, since for any it is
then, if for some , it results in
In particular, all finitely-generated fans are Lipschitz continuous and, if , it results in .
The aforementioned features motivate the choice of fans as a possible tool for approximating more general and less structured set-valued mappings.
In view of the employment of the metric C-increase property in the present approach, the next proposition provides conditions for a fan to be globally metrically C-increasing. Its proof makes use of a well-known order cancellation law, saying that whenever is nonempty, is nonempty convex and bounded, and is nonempty closed and convex, then the following implication holds (see [19, Theorem 3.2.1]):
Let be a fan. If
then H is globally metrically C-increasing and . Conversely, if the fan takes compact values, then condition (2.7) is also necessary for H to be globally metrically C-increasing.
Take arbitrary and . Letting and as in condition (2.7) and setting , one has that and obtains
Conversely, observe first of all that if H takes compact values, then it must be . Indeed, as H is p.h., one has for any , so is a cone, but is the only compact cone. Now, if H is globally metrically C-increasing, for some , taking and , there exists such that
Since it is , by virtue of the order cancellation law, from the last inclusion one obtains
so condition (2.7) is shown to be satisfied with . ∎
(i) Notice that the condition for metric C-increase expressed by (2.7) requires that . As a consequence, whenever working with finitely generated fans, which are supposed to be globally metrically C-increasing, one is forced to assume that .
(ii) Condition (2.7) may be read in terms of strict positivity. Take into account that, with reference to the partial order induced by C, the elements in C are the positive ones. Thus, condition (2.7) postulates the existence of a direction, along which H takes strictly positive values only.
From condition (2.7) one can derive a sufficient condition for the global metric C-increase property, which is specific for fans generated by regular linear mappings. Recall that if is regular (i.e. onto, or equivalently it is an epimorphism), then there exists such that . The quantity
is called exact openness bound of Λ and is used to provide a measure of the regularity (openness or covering) of Λ. For more details on the notion of openness of linear mappings, the reader is referred to [17, Section 1.2.3]. In particular, for exact estimates of , see [17, Corollary 1.58].
Let be a fan generated by . Suppose that
then H is globally metrically C-increasing.
By hypothesis, there exist and such that
Notice that it is possible to assume without loss of generality that , because if it is
then there must exist such that
Since C is a pointed cone, the above inclusions imply , so for every , whence
Furthermore, since is a cone, it is possible to assume that . Letting , since for every , one has
Therefore, it holds
In order to utilize p.h. set-valued mappings and, in particular, fans as an approximation tool for general multi-valued mappings, a concept of differentiation is needed. Among various proposals extending differential calculus to a set-valued context, motivated by the specific features of the subject under study, the notion of prederivative is employed here, as found in . Such a notion has been recently considered for different purposes in the variational analysis literature also in [13, 20].
Definition 2.19 (Prederivative).
Let be a set-valued mapping and let . A p.h. set-valued mapping is said to be a
outer prederivative of F at if for every there exists such that
inner prederivative of F at if for every there exists such that
prederivative of F at if H is both, an outer and an inner prederivative of F at .
It is clear that, whenever a set-valued mapping F happens to be single-valued in a neighborhood of and H is a p.h. mapping, then all cases (i), (ii), and (iii) in Definition 2.19 coincide with the notion of Bouligand derivative (a.k.a. B-derivative), as introduced in . In particular, if , then the above three notions collapse to the notion of Fréchet differentiability for mappings. In full analogy with the calculus for single-valued smooth mappings, in the current context a strict variant of the notion of prederivative, which will be employed in the sequel, may be formulated as follows [13, 20].
Definition 2.20 (Strict prederivative).
Let be a set-valued mapping and let . A p.h. set-valued mapping is said to be a strict prederivative of F at if for every there exists such that
The reader should notice that the notion in Definition 2.19 (ii) and, consequently, the one in Definition 2.19 (iii) are different from the notion of inner T-derivative and of T-derivative, respectively, as proposed in . This happens because the term appears on the left-hand side of the inclusion in Definition 2.19 (ii). Such a choice entails that a strict prederivative in the sense of Definition 2.20 could fail to be a prederivative of the same set-valued mapping. This fact is in contrast with what happens for T-derivative and strict T-derivative, and therefore it causes a shortcoming in the resulting theory. Nevertheless, such a choice seems to be unavoidable in order to obtain the outer tangential approximation of , where the values of H must be included in for (see the proof of Theorem 3.5). In this regard, it could be relevant to observe that in [14, Definition 9.1] (where F is single-valued), the p.h. term appears on the left-hand side of the inclusion defining the inner prederivative.
The next result shows how local approximations expressed by certain prederivatives can be exploited in order to formulate a condition for the metric C-increase property of set-valued mappings around a reference point, relative to a given set.
Proposition 2.22 (Metric C-increase via strict prederivative).
Let be a set-valued mapping, let be a closed set and let . Suppose the following conditions:
F admits a strict prederivative at .
There exist and such that .
H is Lipschitz with constant .
Then F is metrically C-increasing around relative to S with
Let η and u be as in hypothesis (ii). Notice that, without any loss of generality, it is possible to assume that . Moreover, by the positive homogeneity of H, one has
Since holds, according to the definition of Clarke tangent cone, corresponding to there must exist such that for every and for every there is such that . In view of subsequent estimates, it is useful to observe that, by virtue of the Lipschitz continuity of H, it holds
Fix in such a way that
According to hypothesis (i), there exists such that
Now, choose and take arbitrary and .
Since, in particular, and , there is such that . Thus, let us define . By recalling that , it results in
This means that , so it is possible to apply inclusion (2.10), with and . Furthermore, it is also useful to remark that
By inequality (2.11), it is true that . Therefore, since
the last inclusion shows that F is metrically C-increasing around relative to S. The arbitrariness of enables one to get the quantitative estimate of in the thesis. ∎
The condition appearing in hypothesis (ii) can be regarded as a localization of condition (2.7). This shows that the approximation apparatus based on prederivatives transforms properties of approximations into corresponding properties of the mappings to be approximated, as it happens with classical differential calculus and certain specific properties such as metric regularity (see [14, 17]).
3 Main achievements
The main result of the paper, about a tangential approximation of near one of its elements, is established below.
Theorem 3.1 (Inner tangential approximation under C-increase).
With reference to problem (SVI), suppose . Suppose the following conditions:
F is l.s.c. in a neighborhood of .
F is metrically C -increasing around relative to S.
F admits as an outer prederivative at .
Then the following inclusion holds:
If, in addition,
the outer prederivative H of F at is Lipschitz,
then the following stronger inclusion holds:
Take an arbitrary . If , then it is obviously . So, let us suppose henceforth . Observe that, since , and are all cones (remember Remark 2.14 (i)), it is possible to assume without any loss of generality that . According to the characterization of elements in the contingent cone mentioned in Remark 2.9, in order to prove that it suffices to show that
This means that for every and there must exist such that
On the other hand, by virtue of hypothesis (iii), corresponding to ϵ there exists such that
Now, take in such a way that
Since , there exists with the property that . As a consequence of inclusion (3.5), taking into account that , one finds
From the last inclusion, on account of what was recalled in Remark 2.1 (iii), it follows that
Therefore, by exploiting the error bound inequality (2.4), what is possible to do inasmuch as
In order to prove the second inclusion in the thesis, observe first that, since the function
is Lipschitz, it holds
By consequence, in order to show that implies by means of the characterization in (3.3), it suffices to prove the existence of sequences with and with as such that
Again, one can assume that (the case being trivial). Since , there exist with and with such that for every . As a consequence of hypothesis (iv), one finds that for some it must hold
Fix . Correspondingly, by hypothesis (iii) there exists such that the following inclusion holds true:
Take , where