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BY 4.0 license Open Access Published by De Gruyter March 20, 2021

On tangential approximations of the solution set of set-valued inclusions

Amos Uderzo ORCID logo

Abstract

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.

MSC 2010: 49J52; 49J53; 90C30

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 F : n m , a nonempty closed set S n , and a closed, convex and pointed cone C m , with C { 𝟎 } ,

(SVI) find  x S  such that  F ( x ) C .

The solution set to (SVI) will be denoted throughout the paper by

Sol ( SVI ) = { x S : F ( x ) C } .

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 Sol ( SVI ) can be nominally expressed in terms of upper inverse image of C through F, i.e. F + 1 ( C ) = { x n : F ( x ) C } , resulting in Sol ( SVI ) = F + 1 ( C ) S . In spite of such a simple reformulation, Sol ( SVI ) 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 Sol ( SVI ) , 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 Sol ( SVI ) . 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 F + 1 ( C ) S 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 [16]). In its modern formulation, suitable for problems of the form

(1.1) find  x S  such that  f ( x ) C ,

where f : n m is a given single-valued mapping, this theorem states that, under proper regularity assumptions valid at a solution x ¯ f - 1 ( C ) S , the representation

(1.2) T ( f - 1 ( C ) S ; x ¯ ) = D f ( x ¯ ) - 1 ( T ( C ; f ( x ¯ ) ) ) T ( S ; x ¯ )

holds, where D f ( x ¯ ) stands for the strict derivative of f at x ¯ (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 T ( F + 1 ( C ) S ; x ¯ ) . If H : n m acts as a first-order approximation of F near x ¯ , one expects that geometric properties of H result in an easy geometry of the set H + 1 ( C ) . For example, it is well known that if H is positively homogeneous, then H + 1 ( C ) is a cone; if H is concave (in the sense of [26, Definition 2.3]), then H + 1 ( C ) is a convex set. Such correspondences evidently contribute to a better understanding of the local structure of Sol ( SVI ) .

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 x ¯ S is a solution to (SVI) involves all elements in F ( x ¯ ) , 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 F ( x ¯ ) . 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 Sol ( SVI ) . 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 [5]. 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 [9]. Some advances in this direction, already including representations of the contingent cone to Sol ( SVI ) , have been recently obtained in [26], 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 Sol ( SVI ) .

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 Sol ( SVI ) , 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 [15]).

2 Analysis tools

The notation in use throughout the paper is standard: and denote the natural and the real number set, respectively, + m denotes the nonnegative orthant in the Euclidean space m , whose (Euclidean) norm is indicated by | | . The null vector in any Euclidean space is indicated by 𝟎 . Given an element x of a metric space ( X , d ) and a nonnegative real r, B ( x , r ) = { z X : d ( z , x ) r } denotes the closed ball with center x and radius r. In particular, if X = n , then 𝔹 = B ( 𝟎 , 1 ) and 𝕊 = { v 𝔹 : | v | = 1 } stand for the unit ball and the unit sphere, respectively. Given a subset S of an Euclidean space, by int S and bd S the topological interior and the boundary of S are denoted, respectively, whereas co S denotes the convex hull of S. By dist ( x , S ) = inf z S d ( z , x ) the distance of x from a subset S n is denoted, with the convention that dist ( x , ) = + . The r-enlargement of a set S n is indicated by B ( S , r ) = { x n : dist ( x , S ) r } . Given a pair of subsets S 1 , S 2 X , the symbol exc ( S 1 , S 2 ) = sup s S 1 dist ( s , S 2 ) denotes the excess of S 1 over S 2 , where the convention sup x = - is accepted. The symbol Haus ( S 1 , S 2 ) = max { exc ( S 1 , S 2 ) , exc ( S 2 , S 1 ) } indicates the Pompeiu–Hausdorff distance between S 1 and S 2 .

Whenever F : n m is a set-valued mapping,

gph F = { ( x , y ) n × m : y F ( x ) } and dom F = { x n : F ( x ) }

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, ( n ; m ) indicates the space of all linear mappings acting from n to m , endowed with the operator norm . If Λ ( n ; m ) , then Λ denotes the adjoint operator to Λ. Given a function φ : X { ± } , where X is a given set, [ φ 0 ] = φ - 1 ( [ - , 0 ] ) and [ φ > 0 ] = φ - 1 ( ( 0 , + ] ) 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 F : n m is said to be l.s.c. at x ¯ n if for every open set O m such that F ( x ¯ ) O there exists δ O > 0 such that

F ( x ) O for all  x B ( x ¯ , δ O ) .

The mapping F : n m is said to be Lipschitz (continuous) with constant κ 0 if

Haus ( F ( x 1 ) , F ( x 2 ) ) κ | x 1 - x 2 | for all  x 1 , x 2 n .

A known fact which is relevant to the present analysis is that if a set-valued mapping F : n m is l.s.c. at each point of n , then F + 1 ( C ) 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 Sol ( SVI ) = S F + 1 ( C ) is a closed subset (possibly empty) of n .

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 [25]).

Remark 2.1.

Let S m be nonempty and let C m be a closed, convex cone.

  1. If exc ( S , C ) > 0 , for any r > 0 it holds exc ( B ( S , r ) , C ) = exc ( S , C ) + r (additive behavior of the excess with respect to enlargements).

  2. If S m , it holds exc ( S + C , C ) = exc ( S , C ) (invariance of the excess under conic extension).

  3. Let r > 0 . It holds exc ( r 𝔹 , C ) = sup x r 𝔹 inf c C | x - c | sup x r 𝔹 | x | = r .

Remark 2.2.

(i) Given a set-valued mapping F : n m , it is known that the aforementioned semicontinuity property of F implies a corresponding semicontinuity property of the excess function ϕ : n { ± } , which is associated with F and C, namely

(2.1) ϕ ( x ) = exc ( F ( x ) , C ) .

In other words, if F is l.s.c. at x ¯ , then ϕ is l.s.c. at x ¯ (the proof can be found in [25, Lemma 2.3]).

(ii) Since C is a closed set, it is clear that F ( x ) C if and only if ϕ ( x ) 0 . Therefore, the solution set to (SVI) can be characterized, via the function ϕ, in the following terms: Sol ( SVI ) = [ ϕ 0 ] S .

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 [10], given a function φ : X { ± } defined on a metric space ( X , d ) and x ¯ φ - 1 ( ) , the strong slope of φ at x ¯ is defined as the quantity

| φ | ( x ¯ ) = { 0 if  x ¯  is a local minimizer of  φ , lim sup x x ¯ φ ( x ¯ ) - φ ( x ) d ( x , x ¯ ) otherwise.

Notice that, if as a metric space X one takes a closed subset S m containing x ¯ , the above definition becomes

| φ | ( x ¯ ) = { 0 if  x ¯  is a local minimizer of  φ  over  S , inf r > 0 sup x B ( x ¯ , r ) S { x ¯ } φ ( x ¯ ) - φ ( x ) | x - x ¯ | otherwise.

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.

Proposition 2.3.

Let ( X , d ) be a complete metric space, let φ : X [ 0 , + ] be a function l.s.c. on X, and let x ¯ [ φ 0 ] . Suppose that σ > 0 and r > 0 are such that

| φ | ( x ) σ for all  x B ( x ¯ , 2 r ) [ φ > 0 ] .

Then it holds

dist ( x , [ φ 0 ] ) φ ( x ) σ for all  x B ( x ¯ , r ) .

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 C ), i.e. y 1 C y 2 if and only if y 2 - y 1 C .

Definition 2.4 (Metrically C-increasing mapping).

Let S n be a nonempty closed set and let C m be a closed, convex cone, with C { 𝟎 } . Consider a set-valued mapping F : n m :

  1. F is said to be metrically C-increasing around x ¯ dom F relative to S if there exist δ > 0 and α > 1 such that

    (2.2) { for all  x B ( x ¯ , δ ) S  and all  r ( 0 , δ )  there exists  z B ( x , r ) S such that  B ( F ( z ) , α r ) B ( F ( x ) + C , r ) .

    The quantity

    inc C ( F ; S ; x ¯ ) = sup { α > 1 : there exists  δ > 0  for which the inclusion in (2.2) holds }

    is called exact bound of metric C-increase of F around x ¯ , relative to S.

  2. F is said to be globally metrically C-increasing if there exists α > 1 such that

    (2.3) for all  x n  and all  r > 0  there exists  z B ( x , r )  such that  B ( F ( z ) , α r ) B ( F ( x ) + C , r ) .

    The quantity

    inc C ( F ) = sup { α > 1 : the inclusion in (2.3) holds }

    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 κ > 0 such that

inf x B ( x ¯ , r ) φ ( x ) φ ( x ¯ ) - κ r ,

where φ : X { + } is a l.s.c. and bounded from below function defined on a proper (at least, metric) space, x ¯ X is a reference point and r > 0 . 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 [21]). 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 [7]). This led the author to employ the term “metric C-increase” in [25].

Remark 2.5.

(i) Whenever x ¯ int S , the notion of metric C-increase around x ¯ , relative to S, reduces to the notion of local metric C-increase around x ¯ , as defined in [25].

(ii) An equivalent reformulation of the inclusion (2.2) that will be useful is clearly

for all  x B ( x ¯ , δ ) S  and all  r ( 0 , δ )  there exists  z B ( x , r ) S
such that  F ( z ) + α r 𝔹 F ( x ) + C + r 𝔹 .

Example 2.6.

Let F : 2 be defined by

F ( x ) = { y = ( y 1 , y 2 ) 2 : min { y 1 , y 2 } = x }

and let C = + 2 . By a direct check of Definition 2.4 (ii), one can see that the set-valued mapping F is globally metrically + 2 -increasing, with inc C ( F ) = 2 .

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 [25].

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 F : R n R m is a set-valued mapping, C R m is a closed convex cone, and S is a closed set defining a problem (SVI), with x ¯ Sol ( SVI ) . Suppose the following conditions:

  1. F is l.s.c. in B ( x ¯ , δ ) S for some δ > 0 .

  2. F is metrically C -increasing around x ¯ , relatively to S.

Then, for every α ( 1 , inc C ( F ; S ; x ¯ ) ) , there exists δ α > 0 such that

(2.4) dist ( x , Sol ( SVI ) ) exc ( F ( x ) , C ) α - 1 for all  x B ( x ¯ , δ α ) S .

Proof.

Consider the function ϕ : S [ 0 , + ] , defined as in (2.1). According to Remark 2.2 (i), ϕ is l.s.c. on B ( x ¯ , δ ) S by virtue of hypothesis (i). Take α ( 1 , inc C ( F ; S ; x ¯ ) ) . 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 x B ( x ¯ , δ ) S [ ϕ > 0 ] . Then, according to hypothesis (ii), taken any r > 0 such that

r < min { δ , ϕ ( x ) α - 1 } ,

there exists z r B ( x , r ) S such that

(2.5) B ( F ( z r ) , α r ) B ( F ( x ) + C , r ) .

Notice that z r x must hold. This is evident if ϕ ( z r ) = 0 because x [ ϕ > 0 ] . In the case ϕ ( z r ) > 0 , if it were z r = x , on account of inclusion (2.5), by recalling Remark 2.1 (i) and (ii), one would obtain

ϕ ( x ) + α r = exc ( F ( x ) , C ) + α r
= exc ( B ( F ( x ) , α r ) , C )
exc ( B ( F ( x ) + C , r ) , C )
= exc ( F ( x ) , C ) + r
= ϕ ( x ) + r ,

whence α 1 , in contradiction to the assumption on the value of α. Furthermore, again by Remark 2.1 (i) and (ii), and inclusion (2.5), it is possible to observe that

ϕ ( z r ) = exc ( B ( F ( z r ) , α r ) , C ) - α r
exc ( B ( F ( x ) + C , r ) , C ) - α r
= exc ( F ( x ) + C , C ) + r - α r
= ϕ ( x ) + ( 1 - α ) r .

The last inequality chain implies

ϕ ( x ) - ϕ ( z r ) ( α - 1 ) r ( α - 1 ) | x - z r | ,

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

| ϕ | ( x ) = lim sup z 𝑆 x ϕ ( x ) - ϕ ( z ) | x - z |
= inf r > 0 sup z B ( x , r ) S { x } ϕ ( x ) - ϕ ( z ) | x - z |
inf r > 0 ϕ ( x ) - ϕ ( z r ) | x - z r |
α - 1 .

This shows that

| ϕ | ( x ) α - 1 for all  x B ( x ¯ , δ ) S [ ϕ > 0 ] .

Since S, as a closed subset of n , is a complete metric space, Proposition 2.3 guarantees that

dist ( x , Sol ( SVI ) ) = dist ( x , [ ϕ 0 ] S ) ϕ ( x ) α - 1 for all  x B ( x ¯ , δ 2 ) S .

Thus, it suffices to set δ α = δ 2 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 F : defined by

F ( x ) = [ - x 2 , + ) ,

and take S = , C = + and x ¯ = 0 . With these data, the resulting (SVI) evidently admits { 0 } as a solution set. Therefore, one has

dist ( x , Sol ( SVI ) ) = | x | for all  x .

On the other hand, one sees that it is

exc ( F ( x ) , + ) = x 2 for all  x .

As a consequence, for any α > 1 , the error bound inequality

dist ( x , Sol ( SVI ) ) = | x | x 2 α - 1 = exc ( F ( x ) , + ) α - 1

fails to hold in any interval ( - δ α , δ α ) , whatever the value of δ α > 0 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 S n be a nonempty closed set and let x ¯ S . As a first-order approximation of S near x ¯ , the following different cones will be used:

T ( S ; x ¯ ) = { v n : there exist  ( v n ) n  with  v n v  and  ( t n ) n  with  t n 0  such that  x ¯ + t n v n S  for all  n } ,
I ( S ; x ¯ ) = { v n : there exists  δ > 0  such that  x ¯ + t v S  for all  t ( 0 , δ ) } ,
I w ( S ; x ¯ ) = { v n : for all  ϵ > 0  there exists  t ϵ ( 0 , ϵ )  such that  x ¯ + t ϵ v S } ,
T Cl ( S ; x ¯ ) = { v n : for all  ϵ > 0  there exists  τ > 0  such that  for all  x B ( x ¯ , τ )  and all  t ( 0 , τ )
    there exists  v B ( v , ϵ )  such that  x + t v S } .

They are called the contingent cone, the feasible direction cone, the weak feasible direction cone and the Clarke tangent cone to S at x ¯ , 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:

I ( S ; x ¯ ) I w ( S ; x ¯ ) T ( S ; x ¯ ) and T Cl ( S ; x ¯ ) T ( S ; x ¯ ) .

When, in particular, S is locally convex around x ¯ , i.e. there exists r > 0 such that S B ( x ¯ , r ) is convex, then

cl I ( S ; x ¯ ) = cl I w ( S ; x ¯ ) = T ( S ; x ¯ )

(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 r > 0 it is

(2.6) T ( S ; x ¯ ) = T ( S B ( x ¯ , r ) ; x ¯ ) .

Of course, whenever S is a closed convex cone, one finds T ( S ; 𝟎 ) = S .

Remark 2.9.

Given a nonempty S n and x ¯ S , the following characterization of T ( S ; x ¯ ) in terms of the Dini lower derivative of the function x dist ( x , S ) at x ¯ will be useful:

T ( S ; x ¯ ) = { v n : lim inf t 0 dist ( x ¯ + t v , S ) t = 0 }

(see [23, Proposition 11.1.5] and [2], where the above equality actually appears as a definition of the contingent cone to S at x ¯ ).

Given a cone C m , let us recall that the set

C = { v m : v , c 0  for all  c C }

is called (negative) dual cone of C. Whenever S is locally convex around x ¯ (and hence T ( S ; x ¯ ) is convex), such an operator is connected with the normal cone to S at x ¯ in the sense of convex analysis by the following well-known relation:

N ( S ; x ¯ ) = T ( S ; x ¯ ) .

Let φ : n { ± } be a function which is finite around x ¯ n . Following [17], the sets

^ φ ( x ¯ ) = { v n : lim inf x x ¯ φ ( x ) - φ ( x ¯ ) - v , x - x ¯ | x - x ¯ | 0 }

and

^ + φ ( x ¯ ) = { v n : lim sup x x ¯ φ ( x ) - φ ( x ¯ ) - v , x - x ¯ | x - x ¯ | 0 }

are called the Fréchet subdifferential of φ at x ¯ and the Fréchet upper subdifferential of φ at x ¯ , respectively. It is readily seen that, whenever φ is (Fréchet) differentiable at x ¯ , then ^ φ ( x ¯ ) = ^ + φ ( x ¯ ) = { φ ( x ¯ ) } , whereas whenever φ : n is convex (resp. concave), the set ^ φ ( x ¯ ) (resp. ^ + φ ( x ¯ ) ) reduces to the subdifferential (resp. superdifferential) of φ at x ¯ in the sense of convex analysis.

Remark 2.10.

The following variational description of the Fréchet upper subdifferential of φ at x ¯ will be exploited in the sequel: for every v ^ + φ ( x ¯ ) there exists a function σ : n , differentiable at x ¯ and with φ ( x ¯ ) = σ ( x ¯ ) , such that φ ( x ) σ ( x ) for every x n and σ ( x ¯ ) = v (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 H : n m is positively homogeneous (for short, p.h.) if 𝟎 H ( 𝟎 ) and

H ( λ x ) = λ H ( x ) for all  λ > 0  and all  x n .

Within the class of p.h. set-valued mappings, fans will play a prominent role in the present analysis (see [14]).

Definition 2.11 (Fan).

A set-valued mapping H : n m is said to be a fan if it fulfils the following conditions:

  1. It is p.h.

  2. It is convex-valued.

  3. It holds

    H ( x 1 + x 2 ) H ( x 1 ) + H ( x 2 ) for all  x 1 , x 2 n .

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 n and m includes, as a very special case, the space ( n ; m ) . An important class of fan, playing a role in the present analysis, is discussed below.

Example 2.12 (Fans generated by linear mappings).

Let 𝒢 ( n ; m ) be a nonempty, convex and closed set. The set-valued mapping H : n m defined by

H ( x ) = { Λ x : Λ 𝒢 }

is known to be a fan (see [14]). In such a circumstance, the set 𝒢 will be called a generator for H. In particular, whenever 𝒢 is a polytope in ( n ; m ) , the fan generated by 𝒢 will be said to be finitely-generated. For example, in the case m = n , one may take the class of all linear mappings represented by n × n 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 [6]). Note that any finitely-generated fan takes compact values which are polytopes in the range space m . In general, for any fan H generated by linear mappings it must be H ( 𝟎 ) = { 𝟎 } .

Example 2.13.

The set-valued mapping H : , defined by

H ( x ) = { - + if  x < 0 , if  x = 0 , + if  x > 0 ,

is a fan. Since H ( 0 ) = holds, it is clear that H can not be generated by any set 𝒢 ( ; ) .

Further examples of fans are provided in [26].

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 Sol ( SVI ) . In this perspective, the next remark gathers some elementary algebraic/topological properties of such a set.

Remark 2.14.

(i) It is plain to see that if H : n m is a fan and C m is a closed convex cone, then the set H + 1 ( C ) is a convex cone (possibly empty). Notice that, in general, H + 1 ( C ) may happen to be not closed. For example, if taking C = + and such a fan H : as defined in Example 2.13, one finds H + 1 ( C ) = ( 0 , + ) (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 𝒢 ( n ; m ) , it results in

H + 1 ( C ) = Λ 𝒢 Λ - 1 ( C ) .

As an immediate consequence of the last equality, one deduces that the convex cone H + 1 ( C ) is closed whenever H is a fan generated by linear mappings. Furthermore, if a fan H is finitely-generated, i.e.

𝒢 = co { Λ 1 , , Λ p } ,

with Λ i ( n ; m ) , for = 1 , , p , then it results in

H + 1 ( C ) = i = 1 p Λ i - 1 ( C ) .

In this case, each set Λ i - 1 ( C ) turns out to be polyhedral, provided that C is so, and therefore H + 1 ( C ) inherits a polyhedral cone structure.

(iii) Whenever H : n m is a fan generated by a bounded set 𝒢 ( n ; m ) , it turns out to be Lipschitz. More precisely, if l = sup { Λ : Λ 𝒢 } < + , it holds

Haus ( H ( x 1 ) , H ( x 2 ) ) l | x 1 - x 2 | for all  x 1 , x 2 n .

Indeed, since for any y m it is

dist ( y , H ( x 2 ) ) = inf Λ 𝒢 | y - Λ x 2 | ,

then, if y = Λ ~ x 1 for some Λ ~ 𝒢 , it results in

dist ( Λ ~ x 1 , H ( x 2 ) ) = inf Λ 𝒢 | Λ ~ x 1 - Λ x 2 | | Λ ~ x 1 - Λ ~ x 2 | Λ ~ | x 1 - x 2 | .

It follows

exc ( H ( x 1 ) , H ( x 2 ) ) = sup y H ( x 1 ) dist ( y , H ( x 2 ) )
= sup Λ 𝒢 dist ( Λ x 1 , H ( x 2 ) )
sup Λ 𝒢 Λ | x 1 - x 2 |
l | x 1 - x 2 | for all  x 1 , x 2 n .

In particular, all finitely-generated fans are Lipschitz continuous and, if 𝒢 = co { Λ 1 , , Λ p } , it results in l = max i = 1 , , p Λ i .

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 A m is nonempty, B m is nonempty convex and bounded, and C m is nonempty closed and convex, then the following implication holds (see [19, Theorem 3.2.1]):

A + B C + B implies A C .

Proposition 2.15.

Let H : R n R m be a fan. If

(2.7) there exist  η > 0 and  u 𝔹 such that  H ( u ) + η 𝔹 C ,

then H is globally metrically C-increasing and inc C ( H ) η + 1 . Conversely, if the fan H : R n R m takes compact values, then condition (2.7) is also necessary for H to be globally metrically C-increasing.

Proof.

Take arbitrary x n and r > 0 . Letting u 𝔹 and η > 0 as in condition (2.7) and setting z = x + r u , one has that z B ( x , r ) and obtains

H ( z ) + ( η + 1 ) r 𝔹 H ( x ) + r H ( u ) + η r 𝔹 + r 𝔹
= H ( x ) + r ( H ( u ) + η 𝔹 ) + r 𝔹
H ( x ) + C + r 𝔹 .

According to Definition 2.4 (ii) and Remark 2.5, this proves that H is globally metrically C-increasing.

Conversely, observe first of all that if H takes compact values, then it must be H ( 𝟎 ) = { 𝟎 } . Indeed, as H is p.h., one has λ H ( 𝟎 ) = H ( λ 𝟎 ) = H ( 𝟎 ) for any λ > 0 , so H ( 𝟎 ) is a cone, but { 𝟎 } is the only compact cone. Now, if H is globally metrically C-increasing, for some α ( 1 , inc C ( H ) ) , taking x = 𝟎 and r = 1 , there exists v 𝔹 such that

H ( v ) + α 𝔹 H ( 𝟎 ) + C + 𝔹 = C + 𝔹 .

Since it is α 𝔹 = ( α - 1 ) 𝔹 + 𝔹 , by virtue of the order cancellation law, from the last inclusion one obtains

H ( v ) + ( α - 1 ) 𝔹 H ( 𝟎 ) + C ,

so condition (2.7) is shown to be satisfied with η = α - 1 > 0 . ∎

Remark 2.16.

(i) Notice that the condition for metric C-increase expressed by (2.7) requires that int C . As a consequence, whenever working with finitely generated fans, which are supposed to be globally metrically C-increasing, one is forced to assume that int C .

(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.

Example 2.17.

According to Definition 2.4, the fan H considered in Example 2.13 fails to be metrically + -increasing around each point of , relative to S = . Observe that, consistently, condition (2.7) is not satisfied.

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 Λ ( n ; m ) is regular (i.e. onto, or equivalently it is an epimorphism), then there exists η > 0 such that Λ 𝔹 η 𝔹 . The quantity

sur ( Λ ) = sup { η > 0 : Λ 𝔹 η 𝔹 }

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 sur ( Λ ) , see [17, Corollary 1.58].

Corollary 2.18.

Let H : R n R m be a fan generated by G L ( R n ; R m ) . Suppose that

inf Λ 𝒢 sur ( Λ ) > 0 .

If

int ( Λ 𝒢 Λ - 1 ( C ) ) ,

then H is globally metrically C-increasing.

Proof.

By hypothesis, there exist u n and ϵ > 0 such that

u + ϵ 𝔹 Λ 𝒢 Λ - 1 ( C ) .

Notice that it is possible to assume without loss of generality that u 𝟎 , because if it is

𝟎 int ( Λ 𝒢 Λ - 1 ( C ) ) ,

that is,

ϵ 𝔹 Λ 𝒢 Λ - 1 ( C ) ,

then there must exist x 𝟎 such that

Λ x C and Λ ( - x ) C    for all  Λ 𝒢 .

Since C is a pointed cone, the above inclusions imply Λ x = 𝟎 , so x Λ - 1 ( C ) for every Λ 𝒢 , whence

x + ϵ 𝔹 Λ 𝒢 Λ - 1 ( C ) .

Furthermore, since Λ 𝒢 Λ - 1 ( C ) is a cone, it is possible to assume that u 𝔹 . Letting 0 < η < inf Λ 𝒢 sur ( Λ ) , since sur ( Λ ) > η for every Λ 𝒢 , one has

Λ ( ϵ 𝔹 ) ϵ η 𝔹 for all  Λ 𝒢 .

Therefore, it holds

Λ u + ϵ η 𝔹 Λ ( u + ϵ 𝔹 ) C for all  Λ 𝒢 .

According to the definition of H, it follows that H ( u ) + ϵ η 𝔹 C , so the sufficient condition (2.7) for a fan to be globally metrically C-increasing is satisfied. The thesis follows from Proposition 2.15. ∎

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 [14]. Such a notion has been recently considered for different purposes in the variational analysis literature also in [13, 20].

Definition 2.19 (Prederivative).

Let F : n m be a set-valued mapping and let x ¯ dom F . A p.h. set-valued mapping H : n m is said to be a

  1. outer prederivative of F at x ¯ if for every ϵ > 0 there exists δ > 0 such that

    F ( x ) F ( x ¯ ) + H ( x - x ¯ ) + ϵ | x - x ¯ | 𝔹 for all  x B ( x ¯ , δ ) ;

  2. inner prederivative of F at x ¯ if for every ϵ > 0 there exists δ > 0 such that

    F ( x ¯ ) + H ( x - x ¯ ) F ( x ) + ϵ | x - x ¯ | 𝔹 for all  x B ( x ¯ , δ ) ;

  3. prederivative of F at x ¯ if H is both, an outer and an inner prederivative of F at x ¯ .

It is clear that, whenever a set-valued mapping F happens to be single-valued in a neighborhood of x ¯ 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 [22]. In particular, if H ( n ; m ) , 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 F : n m be a set-valued mapping and let x ¯ dom F . A p.h. set-valued mapping H : n m is said to be a strict prederivative of F at x ¯ dom F if for every ϵ > 0 there exists δ > 0 such that

F ( x 2 ) F ( x 1 ) + H ( x 2 - x 1 ) + ϵ | x 2 - x 1 | 𝔹 for all  x 1 , x 2 B ( x ¯ , δ ) .

An articulated discussion on the existence of prederivatives and strict prederivative, on their calculus rules and connections with regularity properties, can be found in [13, 20].

Remark 2.21.

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 [20]. This happens because the term H ( x - x ¯ ) 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 Sol ( SVI ) , where the values of H must be included in T ( C ; y ¯ ) for y ¯ F ( x ¯ ) (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 F : R n R m be a set-valued mapping, let S R n be a closed set and let x ¯ dom F S . Suppose the following conditions:

  1. F admits a strict prederivative H : n m at x ¯ .

  2. There exist η > 0 and u T Cl ( S ; x ¯ ) 𝔹 such that H ( u ) + η 𝔹 C .

  3. H is Lipschitz with constant κ 0 .

Then F is metrically C-increasing around x ¯ relative to S with

inc C ( F ; S ; x ¯ ) η 4 ( κ + 1 ) + 1 .

Proof.

Let η and u be as in hypothesis (ii). Notice that, without any loss of generality, it is possible to assume that η ( 0 , 1 ) . Moreover, by the positive homogeneity of H, one has

(2.8) H ( u 2 ) + η 2 𝔹 = 1 2 [ H ( u ) + η 𝔹 ] 1 2 C = C .

Since u 2 T Cl ( S ; x ¯ ) holds, according to the definition of Clarke tangent cone, corresponding to η / 4 ( κ + 1 ) there must exist τ > 0 such that for every x B ( x ¯ , τ ) and for every t ( 0 , τ ) there is v B ( u 2 , η / 4 ( κ + 1 ) ) such that x + t v S . In view of subsequent estimates, it is useful to observe that, by virtue of the Lipschitz continuity of H, it holds

H ( v ) H ( u 2 ) + κ | v - u 2 | 𝔹
H ( u 2 ) + κ η 4 ( κ + 1 ) 𝔹
(2.9) H ( u 2 ) + η 4 𝔹 .

Fix ϵ > 0 in such a way that

0 < ϵ < min { 1 , η 4 ( κ + 1 ) } .

According to hypothesis (i), there exists δ ϵ > 0 such that

(2.10) F ( x 1 ) F ( x 2 ) + H ( x 1 - x 2 ) + ϵ | x 1 - x 2 | 𝔹 for all  x 1 , x 2 B ( x ¯ , δ ϵ ) .

Now, choose δ * ( 0 , min { τ , δ ϵ / 3 } ) and take arbitrary x B ( x ¯ , δ * ) S and r ( 0 , δ * ) .

Since, in particular, x B ( x ¯ , τ ) and r ( 0 , τ ) , there is v B ( u 2 , η / 4 ( κ + 1 ) ) such that x + r v S . Thus, let us define z = x + r v . By recalling that η < 1 , it results in

| z - x ¯ | | z - x | + | x - x ¯ |
r | v | + δ * < ( | v | + 1 ) δ *
( 1 2 + η 4 ( κ + 1 ) + 1 ) δ *
( 1 2 + 1 4 + 1 ) δ *
< 2 δ *
< δ ϵ .

This means that x , z B ( x ¯ , δ ϵ ) , so it is possible to apply inclusion (2.10), with x 1 = z and x 2 = x . Furthermore, it is also useful to remark that

(2.11) r | v | < r ( 1 2 + η 4 ( κ + 1 ) ) < r ( 1 2 + 1 4 ) < r .

Consequently, by taking into account inclusions (2.8), (2.9) and (2.11), one obtains

F ( z ) + ( η 4 ( κ + 1 ) + 1 - ϵ ) r 𝔹 F ( x ) + r H ( v ) + ϵ r | v | 𝔹 + ( η 4 ( κ + 1 ) + 1 - ϵ ) r 𝔹
F ( x ) + r [ H ( u 2 ) + η 4 𝔹 ] + η r 4 ( κ + 1 ) 𝔹 + [ ϵ r + ( 1 - ϵ ) r ] 𝔹
F ( x ) + r [ H ( u 2 ) + η 2 𝔹 ] + r 𝔹
F ( x ) + r C + r 𝔹
= F ( x ) + C + r 𝔹 .

By inequality (2.11), it is true that z B ( x , r ) S . Therefore, since

η 4 ( κ + 1 ) + 1 - ϵ > 1 ,

the last inclusion shows that F is metrically C-increasing around x ¯ relative to S. The arbitrariness of ϵ > 0 enables one to get the quantitative estimate of inc C ( F ; S ; x ¯ ) 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 Sol ( SVI ) near one of its elements, is established below.

Theorem 3.1 (Inner tangential approximation under C-increase).

With reference to problem (SVI), suppose x ¯ Sol ( SVI ) . Suppose the following conditions:

  1. F is l.s.c. in a neighborhood of x ¯ .

  2. F is metrically C -increasing around x ¯ relative to S.

  3. F admits H : n m as an outer prederivative at x ¯ .

Then the following inclusion holds:

(3.1) H + 1 ( C ) I w ( S ; x ¯ ) T ( Sol ( SVI ) ; x ¯ ) .

If, in addition,

  1. the outer prederivative H of F at x ¯ is Lipschitz,

then the following stronger inclusion holds:

(3.2) H + 1 ( C ) T ( S ; x ¯ ) T ( Sol ( SVI ) ; x ¯ ) .

Proof.

Take an arbitrary v H + 1 ( C ) I w ( S ; x ¯ ) . If v = 𝟎 , then it is obviously v T ( Sol ( SVI ) ; x ¯ ) . So, let us suppose henceforth v 𝟎 . Observe that, since H + 1 ( C ) , I w ( S ; x ¯ ) and T ( Sol ( SVI ) ; x ¯ ) are all cones (remember Remark 2.14 (i)), it is possible to assume without any loss of generality that | v | = 1 . According to the characterization of elements in the contingent cone mentioned in Remark 2.9, in order to prove that v T ( Sol ( SVI ) ; x ¯ ) it suffices to show that

(3.3) lim inf t 0 dist ( x ¯ + t v , Sol ( SVI ) ) t = sup τ > 0 inf t ( 0 , τ ) dist ( x ¯ + t v , Sol ( SVI ) ) t = 0 .

This means that for every τ > 0 and ϵ > 0 there must exist t ( 0 , τ ) such that

(3.4) dist ( x ¯ + t v , Sol ( SVI ) ) t ϵ .

So, fix positive τ and ϵ. According to Lemma 2.7, by virtue of hypotheses (i) and (ii), a local error bound for (SVI) is valid, so corresponding to α ( 1 , inc C ( F ; S ; x ¯ ) ) there exists δ α > 0 such that inequality (2.4) holds.

On the other hand, by virtue of hypothesis (iii), corresponding to ϵ there exists δ ϵ > 0 such that

(3.5) F ( x ) F ( x ¯ ) + H ( x - x ¯ ) + ϵ ( α - 1 ) | x - x ¯ | 𝔹 for all  x B ( x ¯ , δ ϵ ) .

Now, take δ * in such a way that

0 < δ * < min { δ α , δ ϵ , τ } .

Since v I w ( S ; x ¯ ) , there exists t * ( 0 , δ * ) with the property that x ¯ + t * v S B ( x ¯ , δ * ) . As a consequence of inclusion (3.5), taking into account that v H + 1 ( C ) , one finds

F ( x ¯ + t * v ) F ( x ¯ ) + t * H ( v ) + ϵ ( α - 1 ) t * 𝔹
C + t * C + ϵ ( α - 1 ) t * 𝔹
= C + ϵ ( α - 1 ) t * 𝔹 .

From the last inclusion, on account of what was recalled in Remark 2.1 (iii), it follows that

exc ( F ( x ¯ + t * v ) , C ) exc ( C + ϵ ( α - 1 ) t * 𝔹 , C )
= exc ( ϵ ( α - 1 ) t * 𝔹 , C )
ϵ ( α - 1 ) t * .

Therefore, by exploiting the error bound inequality (2.4), what is possible to do inasmuch as

x ¯ + t * v B ( x ¯ , δ α ) S ,

one obtains

dist ( x ¯ + t * v , Sol ( SVI ) ) t * exc ( F ( x ¯ + t * v ) , C ) ( α - 1 ) t * ϵ .

As the last inequality shows that condition (3.4) is satisfied for t = t * ( 0 , τ ) , inclusion (3.1) is proved.

In order to prove the second inclusion in the thesis, observe first that, since the function

x dist ( x , Sol ( SVI ) )

is Lipschitz, it holds

lim inf t 0 dist ( x ¯ + t v , Sol ( SVI ) ) t = lim inf w v t 0 dist ( x ¯ + t w , Sol ( SVI ) ) t .

By consequence, in order to show that v H + 1 ( C ) T ( S ; x ¯ ) implies v T ( Sol ( SVI ) ; x ¯ ) by means of the characterization in (3.3), it suffices to prove the existence of sequences ( v n ) n with v n v and ( t n ) n with t n 0 as n such that

(3.6) lim n dist ( x ¯ + t n v n , Sol ( SVI ) ) t n = 0 .

Again, one can assume that | v | = 1 (the case v = 𝟎 being trivial). Since v H + 1 ( C ) T ( S ; x ¯ ) , there exist ( v n ) n with v n v and ( t n ) n with t n 0 such that x ¯ + t n v n S for every n . As a consequence of hypothesis (iv), one finds that for some κ > 0 it must hold

H ( v n ) H ( v ) + κ | v n - v | 𝔹 for all  n .

Fix ϵ > 0 . Correspondingly, by hypothesis (iii) there exists δ ϵ > 0 such that the following inclusion holds true:

(3.7) F ( x ) F ( x ¯ ) + H ( x - x ¯ ) + | x - x ¯ | ( α - 1 κ + 2 ) ϵ 𝔹 for all  x B ( x ¯ , δ ϵ ) .

Take δ * ( 0 , min { δ α , δ ϵ } ) , where δ α