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A strong metric subregularity analysis of nonsmooth mappings via steepest displacement rate

In this paper, a systematic study of the strong metric subregularity property of mappings is carried out by means of a variational tool, called steepest displacement rate. With the aid of this tool, a simple characterization of strong metric subregularity for multifunctions acting in metric spaces is formulated. The resulting criterion is shown to be useful for establishing stability properties of the strong metric subregularity in the presence of perturbations, as well as for deriving various conditions, enabling to detect such a property in the case of nonsmooth mappings. Some of these conditions, involving several nonsmooth analysis constructions, are then applied in studying the isolated calmness property of the solution mapping to parameterized generalized equations.

play a crucial role. Under this category falls metric regularity, which is most likely the best known and widely employed. Nonetheless, it turns out that metric regularity is not strictly requested in certain circumstances, while its work can be done by a weaker property called metric subregularity, of course at a lower price in terms of problem assumptions. Consider, for instance, the algebraic characterization of the tangent space to a manifold, which is defined by an equation expressed by a smooth mapping. According to a standard argument, this is the key tool for deriving the Euler-Lagrange multiplier rule in nonlinear optimization, often presented among the consequences of the celebrated Lyusternik's theorem (see [1,2,3]). In order to establish the non-trivial inclusion (the kernel of the derivative is contained in the tangent space) metric regularity is usually invoked, even though the mere metric subregularity would be enough. As another example, consider the exact penalization principle for constrained optimization problem with Lipschitz objective function (see [4]). It happens that this principle can be invoked, provided that a certain error bound inequality is valid, and for the latter circumstance the metric subregularity of the constraining mapping is enough. All of this contributed to raise a large interest in metric subregularity, on which a dedicated vast literature does now exist (see [5,6,7,8,9,10] and references therein).
The main drawback of metric subregularity is its lack of robustness under (even small) perturbations. More precisely, it has been observed that such property happens to be broken if adding to a metrically subregular mapping (even single-valued and smooth) a Lipschitz term, yet with a small Lipschitz constant. This well-known phenomenon explains the difficulty in employing perturbation schemes, when studying criteria for detecting metric subregularity. In this paper, a systematic study is proposed of a special variant of metric subregularity, called strong metric subregularity, which is known to exhibit a notable robustness quality, while keeping rather low requirements in comparison with metric regularity. In particular, the present study concentrates on sufficient conditions for the strong metric subregularity of (possibly) nonsmooth mappings. The analysis of this topic is performed by making use of a variational tool called steepest displacement rate, that enables to formulate a general criterion already in a metric space setting. Such an approach leads to a unifying scheme of analysis and emphasizes the connection of the property under study with the notion of local sharp minimality. Strong metric subregularity along with its stability properties and related infinitesimal characterizations have been considered recently by several authors. For an account on various aspects of the emerging theory of strong metric subregularity, the reader may refer to [5].
The contents of the paper are exposed according to the following structure. In Section 2, after the notion of strong metric subregularity is presented and its reformulation in terms of isolated calmness property for the inverse mapping is recalled, the notion of steepest displacement rate is introduced and exploited to establish the basic characterization. The connection with local sharp minimality is also discussed, while several situations arising in different topics are illustrated, aimed at providing motivations for the interest in the main sub-ject of the paper. In Section 3, two relevant manifestations of the robustness behaviour of strong metric subregularity are embedded in the framework of the steepest displacement rate analysis. In Section 4 several known tools of nonsmooth analysis are combined with the main criterion in order to obtain conditions for the strong metric subregularity of nonsmooth mappings. Some of these results are then applied in Section 5 to investigate the isolated calmness property of the solution mapping to parameterized generalized equations, with base and field term. A final section is reserved for general comments on the exposed achievements.

Strong metric subregularity and its equivalent reformulations
Let us start with recalling the main properties under study. This will be done in a metric space setting, which is the natural environment where the Lipschitzian analysis of stability of multifunctions can be conducted. In a metric space (X, d), the distance from a point x ∈ X to a subset S ⊆ X is denoted by dist (x, S), with the convention that dist (x, ∅) = +∞, while B(x, r) denotes the closed ball with center x and radius r.
(ii) A set-valued mapping F : X ⇒ Y between metric spaces is called strongly metrically subregular at (x,ȳ) ∈ graph F if F is metrically subregular at (x,ȳ) and, in addition,x is an isolated point of F −1 (ȳ) or, equivalently, if there exist κ ≥ 0 and r > 0 such that Roughly speaking, whereas the well-known metric regularity property of a mapping F at (x,ȳ) ∈ graph F can be viewed as a quantitative form of local solvability for the inclusion y ∈ F (x), the strong metric subregularity corresponds to a quantitative form of local uniqueness for the solutionx to the particular inclusionȳ ∈ F (x). The independence of these two properties is illustrated in the next example.
Example 1 Let X = Y = R be equipped with its usual Euclidean metric structure. Consider the mapping F 1 : R ⇒ R defined by Clearly, F 1 is strongly metrically subregular at (0, 0), with subreg F 1 (0, 0) = 0, but it fails to be metrically regular near the same point. Notice that F 1 has not closed graph and it is not semicontinuous.
In the same setting, consider the mapping F 2 : R ⇒ R defined by F 2 (x) = [x, +∞). This multifunction is metrically regular near (0, 0), but it is not strongly metrically subregular at the same point.
The basic tool of analysis in use throughout the present section is introduced in the next definition.
Definition 2 (i) Given a function ϕ : X −→ R ∪ {±∞} defined on a metric space and an elementx ∈ dom f , the value (possibly infinite) is called the steepest descent rate of ϕ atx.
(ii) Let F : X ⇒ Y be a set-valued mapping between metric spaces and let (x,ȳ) ∈ graph F . The (nonnegative, possibly infinite) quantity is called the steepest displacement rate of F at (x,ȳ).

Remark 1
The employment of the steepest descent rate in connection with variational problems is witnessed since [11], whereas its application to nondifferentiable optimization goes back at least to [12]. It was with V.F. Demyanov that it became steady exploited for formulating optimality conditions in metric space settings, as a starting point for more involved nonsmooth analysis constructions (see [13,14,15,16,17,18]). The use of the distance function from images of a given multifunction to characterize its Lipschitzian properties follows the spirit of [19].
A first basic characterization of strong metric subregularity is established next as a positivity condition on the steepest displacement rate of a given multifunction.
Proposition 1 A set-valued mapping F : X ⇒ Y is strongly metrically subregular at (x,ȳ) ∈ graph F if and only if Moreover, whenever inequality (3) holds true, it results in with the convention that 1/ + ∞ = 0.
Remark 2 Let us recall that after [20] an elementx ∈ dom ϕ is said to be a local sharp minimizer of a function ϕ : X −→ R ∪ {±∞} defined on a metric space if there exist positive ζ and r such that Clearly,x is a local sharp minimizer of ϕ if and only if ϕ ↓ (x) > 0. Thus, on account of Definition 2 and of Proposition 1, a set-valued mapping F is strongly metrically subregular at (x,ȳ) if and only ifx is a local sharp minimizer of the displacement function x → dist (ȳ, F (x)). Notice that the positivity of the steepest descent rate of a function is a circumstance essentially connected, in more structured settings, with nonsmoothness (see [21]).
Another characterization of the main property under study for a multifunction F : X ⇒ Y can be obtained through the following stability behaviour of its inverse F −1 : Y ⇒ X, i.e. F −1 (y) = {x ∈ X : y ∈ F (x)}.
(ii) A set-valued mapping G : X ⇒ Y between metric spaces is said to have the isolated calmness property at (x,ȳ) ∈ graph G if G is calm at (x,ȳ) and, in addition,ȳ is an isolated point of G(x).
(iii) A function ϕ : Isolated calmness seems to have made its first formal appearance in [22], where it was called "upper-Lipschitz property at a point" (see also [5]).

Theorem 1 ([5])
A set-valued mapping F : X ⇒ Y is strongly metrically subregular at (x,ȳ) ∈ graph F if and only if F −1 has the isolated calmness property at (ȳ,x). In this case, it holds Below several situations, connecting different topics of optimization and variational analysis, are illustrated, where the strong metric subregularity naturally emerges. A further relevant motivation for being interested in strong metric subregularity has to do with the analysis of the solution mapping to generalized equations. This topic will be discussed in Section 5.
Example 2 From Remark 2 it should be clear that every scalar function ϕ : X −→ R ∪ {±∞} defined in a metric space is strongly metrically subregular at each of its local sharp minimizers (if any). As an obvious consequence, the related epigraphical set-valued mapping F ϕ : X ⇒ R, defined as is strongly metrically subregular at (x, ϕ(x)), wheneverx is a local sharp minimizer of ϕ. It is worth noting that if ϕ is calm from below atx, then it can be perturbed in such a way to have the pointx as a local sharp minimizer. Indeed, if for some l > 0 it is ϕ ↓ (x) > −l, then function ϕ + ld(·,x) admits a local sharp minimizer atx. Now, for a lower semicontinuous (henceforth, for short, l.s.c.) proper function ϕ : X −→ R∪{+∞} defined in a complete metric space, the set of all points at which ϕ is calm from below is large enough. In fact, as a direct consequence of the Ekeland variational principle, it is possible to prove that such set is dense in dom ϕ. All of this should show that it is not difficult to generate situations where strong metric subregularity appears.
Example 3 Let L(X, Y) denote the space of all linear bounded operators between two normed spaces, having null vector 0. Given Λ ∈ L(X, Y), its injectivity constant is defined as Λu (see, for instance, [23]). By linearity, for any pair (x,ȳ) ∈ X × Y, withȳ = Λx, one finds While any bounded linear operator Λ is known to be metric subregular at each point of its graph, according to Proposition 1 it is strongly metrically subregular iff α(Λ) > 0 and .
This fact fails to remain true in abstract normed space. Consider, for instance, the identity operator Id : , which is injective, and define for each n ∈ N the elements x n = {x n k } ∈ ℓ 1 as follows It is clear that x n ℓ 1 = 1, whereas x n ℓ ∞ = 1/n for every n ∈ N. Consequently, one has so Id is not strongly metrically subregular (anywhere). On the other hand, it is clear that Id : ℓ 1 −→ ℓ 1 is strongly metrically subregular, with subreg Id(x) = 1, for every x ∈ ℓ 1 . Recall that the injectivity constant is connected with the Banach constant of linear operators, through transposition. Namely, given Λ(X, Y), one defines where Λ ⊤ stands for the transpose of Λ. This allows one to link the notion of strong metric subregularity with that of metric regularity, which is well known to amount to openness (at a linear rate) in the case of linear operators. More precisely, whenever Λ is open, one has 0 < β(Λ) = α(Λ ⊤ ), so that Λ ⊤ is strongly metrically subregular. For more details see [23]. The current example helps also to illustrate the fact that, whereas the appearance of sharp minimality for a given function is a syntom of nonsmoothness, strong metric subregularity is a property that may happen to take place for very nice (even linear) mappings.
Example 4 Let ϕ : X −→ R ∪ {+∞} be a proper, l.s.c. convex function defined on a Banach space X, whose dual is indicated by X * . Let us denote by ∂ϕ(x) the subdifferential of ϕ atx ∈ dom ϕ in the sense of convex analysis. Generalizing a previous result valid in Hilbert spaces, in [24] it is has been proved that the set-valued mapping ∂ϕ : X ⇒ X * is strongly metrically subregular at (x,x * ) ∈ graph ∂ϕ if and only if there exist positive γ and r such that where ·, · : X * × X −→ R denotes the duality pairing X * and X. In particular, in the case of a (global) minimizerx of ϕ, ∂ϕ is strongly metrically subregular at (x, 0 * ), where 0 * stands for the null vector of X * , iff The last inequality formalizes a variational behaviour known as quadratic growth condition, which has been studied in connection with second-order sufficient conditions in nonlinear programming (see [25]). Notice that, if a function admits a sharp minimizer, it satisfies the quadratic growth condition around that point, but the converse may not be true. Similar characterizations of various metric regularity properties have been recently extended to the Mordukhovich subdifferential mapping (see [26]). Investigations by means of second-order variational analysis tools revealed that they are also interrelated to the tilt-stability of local minimizer (see [26,27]).
Remark 3 From Example 4 it is possible to see at once that if ϕ is a proper, l.s.c. convex function, whose subdifferential mapping is strongly metrically subregular at (x, 0 * ), wherex ∈ dom ϕ is a minimizer of ϕ, thenx turns out to be Tykhonov well-posed, namely every minimizing sequence {x n } of ϕ converges tox. Now, it is worth noting that the notion of strong metric subregularity generalizes, yet in a local form, such a behaviour to solutions of equations/inclusions. More precisely, if a set-valued mapping F : X ⇒ Y , defining withȳ the inclusionȳ ∈ F (x), is strongly metrically subregular at (x,ȳ), then for every sequence {y n } in Y , with y n −→ȳ as n → ∞, and for every sequence {x n } in X of solutions of the inclusions with data perturbed y n ∈ F (x), according to (2) one finds d(x n ,x) ≤ κdist (ȳ, F (x n )) ≤ κd(ȳ, y n ), so x n −→x, provided that the elements of {x n } fall in a proper neighbourhood ofx.

Perturbation stability
As it happens for other Lipschitzian properties of multifunctions, a method for establishing criteria or conditions for the validity of strong metric subregularity consists in analyzing its stability in the presence of perturbations. Two results of this type are presented in what follows, which are both proved through the criterion discussed in Section 2.
Theorem 2 Let F : X ⇒ Y be a set-valued mapping between metric spaces and let g : Z −→ X a mapping defined on a metric space. Letz ∈ Z and let (g(z),ȳ) ∈ graph F . Suppose that: (i) g is continuous atz and strongly metrically subregular atz; (ii) F is strongly metrically subregular at (g(z),ȳ). Then, their composition F • g : Z ⇒ Y is strongly metrically subregular at (z,ȳ), and it results in Proof Setx = g(z). Since F is strongly metrically subregular at (x,ȳ), corresponding to an arbitrary κ F > subreg F (x,ȳ), there exists r > 0 such that Owing to hypothesis (i), corresponding to an arbitrary κ g > subreg g(x), there exists δ > 0 such that, up to a further reduction of its value, if neeeded, From inequalities (10) and (9), one obtains To complete the proof it remains to apply Proposition 1.
The following example shows that the continuity assumption on the inner mapping can not be dropped out, in general.
Herez =x =ȳ = 0. It is evident that both g and F have a sharp minimizer at 0, whereas their composition F • g ≡ 0 does not.
Strong metric subregularity is not preserved under composition of setvalued mappings, as shown by the next counterexample.

Example 6
In the same setting as in Example 5, let G : R ⇒ R be defined by and let F be as in the previous example. As one readily checks, G is strongly metrically subregular at (0, 0). If composing G and F , one finds otherwise.
It is easily seen that F • G fails to be strongly metrically subregular at (0, 0). Notice that multifunction G is upper hemicontinuous at 0.
For the next result, a slightly more structured setting is needed.
Theorem 3 Let F : X ⇒ Y be a set-valued mapping defined on a metric space X and taking values in a linear metric space Y , whose metric is shift invariant. If F is strongly metrically subregular at (x,ȳ) ∈ graph F , then for any mapping g : X −→ Y such that subreg F (x,ȳ) · clm g(x) < 1, then the set-valued mapping F + g is strongly metrically subregular at (x,ȳ + g(x)) and it results in .
Proof Notice that, by virtue of the shift-invariance of the metric on Y , for any ).

Consequently, one obtains
The strong metric subregularity of F + g at (x,ȳ + g(x)) follows at once by the characterization provided in Proposition 1, whereas the estimate (11) is a straightforward consequence of (4).
Remark 4 (i) The result provided by Theorem 3 on the persistence of strong metric subregularity under calm additive perturbations can be found in [5] (see Theorem 3I.6), formulated for multifunctions acting in finite-dimensional spaces, with a different proof. It is worth noting that, since neither the Ekeland variational principle nor the convergence of iteration procedures are used in the proof of Theorem 3, metric completeness plays no role in the above robustness phenomenon. Instead, it may be viewed as a direct consequence of a stability behaviour for the local sharp minimality called superstability, which was observed already by B.T. Polyak (see [20]). Essentially, it means that a point preserves its local minimality even in the presence of additive calm perturbations.
This makes the robustness of strong metric subregularity different from the corresponding behaviour of metric regularity, requiring on one hand metric completeness and on the other hand the Lipschitz property of perturbations (see [5,28]).
(ii) Note that the shift-invariance assumption on the metric of Y is not actually restrictive. Indeed, a result due to Kakutani ensures that any linear metric space can be equivalently remetrized by a shift-invariant metric (see Theorem 2.2.11 in [29]).

Strong metric subregularity of nonsmooth mappings
The main subject of this paper is the strong metric subregularity of (possibly) nonsmooth mappings f : X −→ Y. To deal with them, throughout this section (X, · ) and (Y, · ) are supposed to be normed (vector) spaces. The (closed) unit ball and the unit sphere in any normed space are indicated by B and S, respectively, whereas, in the case of dual spaces, by B * and S * , respectively.

A criterium via first-order ǫ-approximations
Differentiability is a wise combination of linearity and approximation. The approach of analysis considered in this subsection relies on the employment of positively homogeneous (for short, p.h.) mappings as an appealing substitute of derivatives (that are linear operators), while calmness replaces the classical convergence of the remainder term. To do so, set H(X, Y) = {h : X −→ Y : p.h. and continuous at 0}.
Definition 4 Let f : X −→ Y be a mapping between normed spaces, let x ∈ X and let ǫ > 0. A mapping h ∈ H(X, Y) is said to be a first-order ǫ-approximation of f atx if clm (f − h(· −x))(x) < ǫ.
Remark 5 Whenever h is a first-order ǫ-approximation of f atx, the mapping f (x) + h(· −x) is a special case of what is called in [5] an "estimator". Of course, first-order ǫ-approximation is a nonsmooth analysis notion, which allows to include (Fréchet) differentiability. Indeed, note that if f is Fréchet differentiable atx, with derivative Df (x) ∈ L(X, Y), then Df (x) is a first-order ǫ-approximation of f atx, for every ǫ > 0. P.h. functions and mappings, or some special classes of them, have been utilized as a rough material for constructing generalized derivatives since the very birth of nonsmooth analysis (see, for instance, [30,31]). On the other hand, the idea of studying properties of nonlinear mappings by means of "approximate differentials", which avoid differentiability assumptions, precedes even nonsmooth analysis (see, for instance, [32]).
After having replaced linear operators with p.h. mappings, the next step consists in extending to H(X, Y) the definition of injective constant, by letting Remark 6 By applying Proposition 1 it is readily seen that h ∈ H(X, Y) is strongly metrically subregular at 0 if and only if it holds α 0 (h) > 0. It is to be noted however that, in contrast with the linear case, such a characterization is not valid for the strong metric subregularity of h at every point of X. Consider, for instance, the norm function · : X −→ R, with the dimension of X being greater than 1 (possibly infinite). Clearly, it is α 0 ( · ) = 1. Taking any element u in the unit sphere S of X, one finds and hence · fails to be strongly metrically subregular atū, even though it is so at 0.
Theorem 4 Let f : X −→ Y be a mapping between normed spaces and let x ∈ X. If f is first-order ǫ-approximated atx by h ∈ H(X, Y), with α 0 (h) > ǫ, then f is strongly metrically subregular atx, and Vice versa, if f is strongly metrically subregular atx, then for any mapping so h is strongly metrically subregular at 0.
Proof According to Definition 4, for both the assertions of the thesis the respective hypotheses imply To prove the first assertion it suffices to apply Theorem 3, with F anf g given by respectively, and to observe that F is strongly metrically subregular atx iff h is so at 0, while clm g(x) = clm (f − h(· −x))(x). Analogoulsy, to prove the second assertion, it suffices to apply once again Theorem 3, now with Indeed, f is strongly metrically subregular atx iff F is so at 0, whereas The quantitative estimates complementing the thesis are direct consequences of inequality (11).
As a special case of Theorem 4 it is possible to derive the following criterion for smooth mappings.
In particular, if X and Y are finite-dimensional spaces, f is strongly metrically subregular atx if and only if

A sufficient condition via outer ǫ-prederivative
In order to introduce the next tool of nonsmooth analysis to be used, recall that a set-valued mapping F : X ⇒ Y is said to be p.h. if 0 ∈ F (0) and F (tx) = tF (x) for all x ∈ X and t > 0. In [33] p.h. set-valued mappings have been used to define a notion of generalized derivative. Below, a generalization of it, which seems to be adequate for the purposes of the present analysis, is introduced.
Definition 5 Let f : X −→ Y be a mapping between normed spaces, let x ∈ X and let A : X ⇒ Y a p.h. homogeneous set-valued mapping. Given ǫ > 0, A is said to be an outer (Fréchet) ǫ-prederivative of f atx if there exists δ > 0 and a function r : δB −→ [0, ǫ] such that Remark 7 Recall that, according to [33], A is said to be an outer prederivative of f atx if (12) holds true with a function r : δB −→ [0, ǫ] such that lim v→0 r(v) = 0. In such an event, A is an outer ǫ-prederivative of f atx, for every ǫ > 0.
Given a p.h. set-valued mapping A : X ⇒ Y, to detect its strong metric subregularity at (0, 0), it seems to be natural to introduce a injectivity constant notion as follows In the light of Prosition 1, it is readily seen that A is strongly metrically subregular at (0, 0) if and only α(A) > 0 and, upon this condition, it is By employing the above nonsmooth analysis tools, one can establish the following sufficient condition for strong metric subregularity.
Theorem 5 Suppose that a mapping f : X −→ Y between normed spaces admits an outer ǫ-prederivative A : X ⇒ Y atx, such that α(A) > ǫ. Then, f is strongly metrically subregular atx and If, in particular, A is an outer prederivative of f atx, then then stricter estimate holds .
Proof Let positive ǫ, δ and r : δB −→ [0, ǫ] be as in Definition 5. Then one obtains for every x ∈ B(x, δ) On the other hand, observe that for any v ∈ δB\{0} it is Therefore, lettingȳ = f (x), from inequality (13) it follows Thus, to show the first assertion in the thesis, it suffices to apply the characterization stated in Proposition 1, along with the estimate (4). For the second assertion, on account of Remark 7 and of the last inequalities, one has that |f | ↓ (x,ȳ) ≥ α(A) − ǫ, ∀ǫ ∈ (0, α(A)).
which leads immediately to the estimate to be proved.
Remark 8 Among the p.h. set-valued mappings that can be used as prederivatives, one can consider in particular those generated by a convex weakly closed set of linear operators. In other terms, given a set U ⊆ L(X, Y) convex and closed with respect to the weak topology, let According to [33], this is an example of fan. In this case one has Notice that, whenever f is Fréchet differentiable atx, Definition 5 applies with Therefore the sufficient part of Corollary 1 can be achieved also from Theorem 5.

A scalarization approach
Let (Y, · ) be now a normed space, which is partially ordered by a relation ≤ Y or, equivalently, by a convex cone Y + ⊆ Y, in the sense that In this setting, a mapping f : Y + -convex mappings are found quite easily in nature. For instance, if f : Since the scalarization approach to strong metric subregularity exploits the connection of that property with the sharp minimality of scalarized terms, the following characterization of a sharp minimizer for a convex function will be useful in the sequel. Proof Necessity: suppose pointx to be a sharp minimizer of ϕ and take ǫ ∈ (0, ϕ ↓ (x)). Then, setting ρ = ϕ ↓ (x) − ǫ one gets This means that ρB * ⊆ ∂ϕ(x) and, by arbitrariness of ǫ, that ϕ ↓ (x) ≤ sup{ρ > 0 : ρB * ⊆ ∂ϕ(x)}. Sufficiency: from the definition of sugradient of ϕ atx, it is possible to deduce Since by hypothesis there exists ρ > 0 such that ρB * ⊆ ∂ϕ(x), it is true that x * , u = ρ, ∀u ∈ X : u = 1.
From this and the previous inequality it is possible to conclude that Actually, this shows that ϕ ↓ (x) ≥ sup{ρ > 0 : ρB * ⊆ ∂ϕ(x)}. The proof is complete.
To formulate the next condition for the strong metric subregularity of a Y + -convex mapping f : Theorem 6 Let f : X −→ Y be a mapping between normed spaces, with Y partially ordered by a cone Y + , and letx ∈ X. Suppose that f is Y + -convex and 0 * ∈ Then f is strongly metrically subregular atx. Moreover, one has .
Proof From the well-known dual representation of a norm v = sup Since by hypothesis there exist ρ > 0 and y * 0 ∈ S * ∩ Y * + such that ρB * ⊆ ∂(y * 0 • f )(x) and function y * 0 • f is convex, then in the light of Lemma 1,x is a sharp minimizer of y * 0 • f . Consequently, settingȳ = f (x), from the last inequality one has The proof of all assertions in the thesis is therefore completed by applying Proposition 1.
Theorem 6 demonstrates a typical use of the scalarization method in the presence of convexity assumption. It should be clear that this method extends its potential far beyond convexity and can be employed in combination with more general subdifferential constructions. For example, by utilizing the Fréchet subdifferential, defined as (for more details, the reader is referred to [2,28,34,35]), the following finitedimensional generalization of Lemma 1 has been proved in [21] (see Theorem 4 therein).
The above lemma allows one to obtain the next result valid for mappings defined in a finite-dimensional space.

Theorem 7
Given a mapping f : then f is strongly metrically subregular atx.
Proof The thesis can be achieved through the same argument as in the proof of Theorem 6. Indeed, by hypothesis there exists y * 0 ∈ S * such that 0 * ∈ int ∂(y * 0 • f )(x). In the light of Lemma 2 this implies Since it is by virtue of inequality (14) it follows Proposition 1 allows one to complete the proof.

An application
An application of the above exposed ideas and techniques is going to be illustrated now, which concerns the stability behaviour of solution mappings to generalized equations. A generalized equation is a rather general problem that is able to provide a proper framework for studying several specific issues in mathematical analysis, having or not having a variational nature. Among the others, let us mention optimality conditions in constrained or unconstrained optimization, various types of constraint systems, variational inequalities and complementarity problems, equilibrium problems, differential inclusions.
Here, parameterized generalized equations are considered that can be formalized as follows where f : P × X −→ Y (sometimes referred to as the base of (GE)) and T : X ⇒ Y (referred to as the field) are the problem data. (P, d) is a metric space where the parameters vary, whereas (X, · ) and (Y, · ) are supposed to be normed vector spaces. The solution mapping associated to (GE) is the (generally) set-valued mapping implicitly defined by In this context, an issue of interest is how to certify and to quantify a certain stability behaviour of S f,T , near a solutionx ∈ S f,T (p). More precisely, here the stability behaviour quantitatively described by the isolated calmness property is investigated. This amounts to establish for solutions to (GE) lying near a reference one a reaction, which is (directly) proportional to the parameter variations. Following the spirit of classical and more recent implicit function theorems, this question is approached by analyzing a semplified variant of (GE), called approximated generalized equation (AGE), on which the main regularity assumption is made. Of course, a (AGE) can be defined in several ways, depending on the features of the problem data. In what follows, dealing with a nonsmooth analysis setting, the use of an adaptation of the outer ǫ-prederivative is proposed.
Definition 6 Given ǫ > 0, a p.h. set-valued mapping A : X ⇒ Y is said to be a partial outer ǫ-prederivative of a mapping f : P × X −→ Y at (p,x), uniformly with respect to p, if there exist positive δ and ζ and a function r : Now, assuming that the base f of (GE) admits, for some ǫ > 0, as a partial outer ǫ-prederivative at (p,x) a mapping A, one can associate with (GE) an approximated generalized equation defined by It turns out that the strong metric subregularity of the mapping A + T is a key assumption to guarantee the isolated calmness property of S f,T , as below stated.
Theorem 8 With reference to a generalized equation (GE), letx ∈ S f,T (p). Suppose the data of (GE) to satisfy the following assumptions: (i) function f (·,x) is calm atp with modulus clm f (·,x)(p); (ii) f admits a partial outer ǫ-prederivative A at (p,x), uniformly with respect to p; (iii) the set-valued mapping x ⇒ f (p,x)+A(x−x)+T (x) is strongly metrically subregular at (x, 0), with modulus subreg (A + T )(x, 0), such that Then, S f,T has the isolated calmness property at (p,x) and the below estimate holds Proof Take an arbitrary η such that what is possible by virtue of condition (15). By the assumption (i), corresponding to η there exists ζ > 0 such that By the assumption (ii), there existζ ∈ (0, ζ), δ > 0 and a function r : Consequently, one obtains for every x ∈ B(x, δ) and p ∈ B(p,ζ), wherefrom it follows Now, according to assumption (iii), sincex is evidently a solution to (AGE), corresponding to η > 0 there existsδ ∈ (0, δ) such that for every x ∈ B(x,δ) and p ∈ B(p,ζ). As a consequence, whenever it is x ∈ S f,T (p) ∩ B(x,δ), it results in for every p ∈ B(p,ζ). The last inequality shows that S f,T has the isolated calmness property at (p,x) with From the last inequality and the arbitrariness of η, it is possible to deduce the estimate in the thesis, thereby completing the proof.
Corollary 2 With reference to a generalized equation (GE), letx ∈ S f,T (p). Suppose the data of (GE) to satisfy the following assumptions: (i) function f (·,x) is calm atp with modulus clm f (·,x)(p); (ii') f has a partial outer prederivative A at (p,x), uniformly with respect to p; (iii) the set-valued mapping x ⇒ f (p,x)+A(x−x)+T (x) is strongly metrically subregular at (x, 0), with modulus subreg (A + T )(x, 0). Then S f,T has the isolated calmness property at (p,x) and the stricter estimate holds.
Proof The thesis can be easily achieved by applying Theorem 8. Recall indeed that with assumption (ii') being valid, A is an outer partial ǫ-prederivative of f at (p,x), for any ǫ > 0. Then, it suffices to observe that, since ǫ can be taken arbitrarily "small", condition (15) is fulfilled independently of the value of subreg (A + T )(x, 0).

Remark 9
A result quite close to Theorem 8, called "implicit mapping theorem with strong metric subregularity", can be found in [5] (Theorem 3I.12). Instead of prederivatives, partial estimators (i.e. first-order ǫ-approximations) are employed there. In this regard, it is must be noted that the technique of proof in Theorem 8 can be readily adapted to derive a version of it, employing partial first-order ǫ-approximations of the base term.
Theorem 8 reduces the study of the isolated calmness property of S f,T to the certification of the strong metric subregularity of the set-valued mapping defining (AGE). The latter question is expected to be easier to be faced than a direct study of S f,T , inasmuch as the former set-valued mapping is explicitly defined in terms of problem data or their approximation, while S f,T can be hardly calculated in practice. Besides, in some special case, the study of the strong metric subregularity of the set-valued mapping defining (AGE) may happen to be particularly simple. Let us consider, as an example, the case in which the field term T happens to be single-valued nearx.

Corollary 3
Letx ∈ S f,T (p) be a solution to (GE). Suppose that: (i) function f (·,x) is calm atp with modulus clm f (·,x)(p); (ii) f admits a partial outer ǫ-prederivative A at (p,x), uniformly with respect to p; (iii) T is single-valued nearx and calm atx, with modulus clm T (x); (iv) the following condition holds Then S f,T has the isolated calmness property at (p,x) and the following modulus estimate holds clm S f,T (p,x) ≤ clm f (·,x)(p) α(A) − clm T (x) − ǫ .
Consequently, the set-valued mapping x ⇒ f (p,x) + A(x −x) + T (x) turns out to be strongly metrically subregular at (x, 0), with .
The last inequality shows that S f,T fulfils the isolated calmness property at (p,x) and, by arbitrariness of η, it allows one to achieve the asserted modulus estimation.

Conclusions
The appraoch of analysis proposed in this paper shows that several techniques for detecting strong metric subregularity of nonsmooth mappings can be derived from a unique elementary criterion, based on the notion of steepest displacement rate, which can be formulated already in a metric space setting. This criterion, besides providing a unifying scheme of analysis with transparent proofs, emphasizes the variational nature of the property under study. Optimization (especially, nondifferentiable optimization) is well recognized as a field where many results and constructions of set-valued analysis are fruitfully applied. The findings of the present study should contribue to the make it evident that, simmetrically, nondifferentiable optimization can provide useful insights and methods for investigating properties of multifunctions, some of them not necessarily related to extremum problems. This seems to agree with the very spirit of the Euler's variational faith.