Metric subregularity of multifunctions: First and

INFORMS

MATHEMATICS OF OPERATIONS RESEARCH Vol. 00, No. 0, Xxxxx 0000, pp. 000–000 issn 0364-765X | eissn 1526-5471 | 00 | 0000 | 0001

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Metric subregularity of multifunctions: First and second order infinitesimal characterizations* Huynh Van Ngai Department of Mathematics, University of Quy Nhon, 170 An Duong Vuong, Quy Nhon, Viet Nam [email protected], [email protected]

Phan Nhat Tinh Department of Mathematics, Hue University of Sciences, 77 Nguyen Hue, Hue, Vietnam [email protected]

Metric subregularity/regularity of multifunctions are fundamental notions in variational analysis and optimization. Using the concept of strong slope, we firstly in this paper establish a criterion for metric subregularity of multifunctions between metric spaces. Next, we use a combination of abstract coderivatives and contingent derivatives to derive verifiable first order conditions ensuring metric subregularity of multifunctions between Banach spaces. Then, using second order approximations of convex multifunctions, we establish a second order condition for metric subregularity of mixed smooth-convex constraint systems, which generalizes a result established recently by Gfrerer in [25]. Key words : Error bound, Metric subregularity, Coderivative, Contingent derivative . MSC2000 subject classification : 47H04, 49J53, 90C31. OR/MS subject classification : Primary: Variational Analysis ; secondary: Optimisation

1. Introduction Many problems arising in variational analysis, optimization, variational inequalities and other areas in mathematics leads to generalized equations, that is the problem of finding a solution to an equation of the form y¯ ∈ F (x),

(1)

where y¯ ∈ Y, F : X ⇒ Y is a multifunction, X and Y are metric spaces. A typical illustration of encountered problems is a constraint system of inequalities/equalities. Indeed, if we consider a mapping f : Rn → Rk+d with f (x) = (f1 (x), · · · , fk (x), fk+1 (x), · · · , fk+d (x)), and if we define F (x) = f (x) + Rk+ × {0}d , then system (S ) consisting of those points x for which fi (x) ≤ 0, i ∈ {1, · · · , k }, fi (x) = 0, i ∈ {k + 1, · · · , k + d},

(2)

* This research was supported by the VIASM (Vietnam Institute of Avanced Study on Mathematics) and the research of the second author is partially supported by the NAFOSTED of Vietnam 1

Huynh and Phan: Metric subregularity of multifunctions c 0000 INFORMS Mathematics of Operations Research 00(0), pp. 000–000, ⃝

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can be rewritten as (1). It is now well established that a central issue in variational analysis and in optimization is to study the behavior of the solution set of equation (1). A key to this is the concept of metric regularity. Let us first recall this important notion. Let F : X ⇒ Y be a multifunction between metric spaces X and Y supposed to be endowed with metrics both denoted by d(·, ·). A mapping F is said to be metrically regular at some x ¯ ∈ X with respect to y¯ ∈ F (¯ x) if there exist τ > 0 and a neighborhood U × V of (¯ x, y¯) such that d(x, F −1 (y)) ≤ τ d(y, F (x)) for all

(x, y) ∈ U × V.

(3)

In the previous inequality we use the standard notation d(x, C) = inf z∈C d(x, z), with the convention ¯ r) the that d(x, C) = +∞ whenever C is empty. In what follows, we will denote B(x, r) and B(x, open and closed balls centered at x with radius r. The concept of metric regularity goes back to the surjectivity of a linear continuous mapping in the Banach open mapping theorem and to its extension to nonlinear operators known as the Lyusternik & Graves Theorem ([42], [24], see also [14] and [17]). The reader is for instance referred to the works by [5, 6, 7, 9, 8, 13, 27, 28, 33, 34, 44, 45, 43, 46, 51, 52] and the references given therein for many theoretical results on the metric regularity as well as its various applications. A weaker property of metric regularity, called metric subregularity, where the inequality (3) holds only for fixed y¯, i.e., the set-valued mapping F is said to be metrically subregular at x ¯ with respect to y¯ ∈ F (¯ x) (or, shortly at (¯ x, y¯) ∈ gph F ) if there exist τ > 0 and a neighborhood U of x ¯ such that d(x, F −1 (¯ y )) ≤ τ d(¯ y , F (x)) for all x ∈ U.

(4)

When F is metrically subregular at (¯ x, y¯) and F −1 (¯ y ) = {x ¯}, we say that F is strongly metrically subregular at (¯ x, y¯). The metric subregularity of F at (¯ x, y¯) is equivalent to calmness of the inverse mapping F −1 : Y ⇒ X at (¯ y, x ¯) in the following sense: there exist τ > 0 and a neighborhood U × V of (¯ x, y¯) such that e(F −1 (y) ∩ U, F −1 (¯ y )) ≤ τ d(y, y¯), y ∈ V where e(Q, P ) denotes the excess of Q over P, e(Q, P ) := inf {ε > 0 : Q ⊆ B(P, ε)},

B(P, ε) := {x ∈ X : d(x, P ) < ε}.

The metric subregularity, which is closely related to the error bound property, plays a crucial role in the study of constrained optimization problems for which the constraints are formulated as an inclusion associated to a set-valued mapping F : 0 ∈ F (x). This property has found a large range of applications in different areas of variational analysis as well as in optimization: for example in the theory of optimality conditions, in the theory of subdifferentiability, in penalty methods in mathematical progamming; in the convergence analysis of algorithms for solving equations or inclusions (see, e.g., [12, 29, 30, 36, 50, 60, 41]). In the literature, it was established several sufficient conditions in terms of some coderivatives or contingent derivatives for metric subregularity of multifunctions (see, for instance, [5, 41, 49, 48, 60]). Recently, in [25], Gfrerer has established a first order point-based criteria for the metric subregularity of set-valued mappings. This characterization is based on the so-called limit set critical for metric subregularity. Then, this author has given a second order characterization for smooth constraint systems of the form: 0 ∈ g(x) − C,

(5)

where, g : X → Y is a C 1 − mapping (that is, g is continuously differentiable) and C ⊆ Y is a closed convex set.

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The first purpose of this paper is to establish a new criterion for metric subregularity of multifunctions acting on metric spaces. This criterion is given in terms of the strong slope of the lower semicontinuous envelope of the distance function associated to the set-valued mapping under consideration. We also show that this criterion is almost necessary in the sense that if this sufficient condition is not verified, then we can find a Lipschitz mapping h with the Lipschitz modulus 0 at x ¯ such that F + h is not metrically subregular at (¯ x, y¯). Based on this characterization, we then give a point-based first order sufficient condition based on a combination of abstract coderivatives and contingent derivatives for metric subregularity. This condition is sharper than the one established in [25]. We also show that, for mixed smooth-convex constraint systems: 0 ∈ g(x) − F (x),

(6)

where, g : X → Y is a C 1 −mapping and F : X ⇒ Y is a closed convex multifunction (that is, the graph of F is closed and convex), this sufficient condition is actually a necessary condition. Secondly, we give a second order condition for metric subregularity of mixed smooth-convex system (6). This second order condition is based on second order approximations of convex multifunctions, and it generalizes the one given in [25]. 2. Metric subregularity on metric spaces Let X, Y be metric spaces and let F : X ⇒ Y be a closed multifunction between X and Y. For a given pair (¯ x, y¯) ∈ gph F , we consider the following inclusion : Find x ∈ X such that y¯ ∈ F (x). (7) Denote by S the solution set of inclusion (7), i.e., S := F −1 (¯ y ) = {x ∈ X :

y¯ ∈ F (x)}.

(8)

Instead of using the distance function from y¯ to F (x) which is not in general lower semicontinuous, we will use its lower semicontinuous envelope φ : X → R ∪ {+∞} defined by φ(x) := lim inf d(¯ y , F (u)), u→x

x ∈ X.

(9)

Obviously, when F is a closed multifunction, one has S = {x ∈ X : φ(x) = 0}, and metric subregularity of F at (¯ x, y¯) is equivalent to the error bound property of the function φ at x ¯, that is, there exist τ > 0 and δ > 0 such that d(x, S) ≤ τ φ(x) for all x ∈ B(¯ x, δ).

(10)

Recall from De Giorgi, Marino & Tosques [15], that the strong slope |∇|φ(x) of the function φ at x ∈ Dom φ := {x ∈ X : φ(x) < +∞} is the quantity defined by |∇|φ(x) = 0 if x is a local minimizer of φ, and φ(x) − φ(y) , |∇|φ(x) = lim sup d(x, y) y→x,y̸=x otherwise. For x ∈ / Dom φ, we set |∇|φ(x) = +∞. For a locally Lipschitz mapping h : X → Y at x ¯ ∈ X, we use the notation lip (h, x ¯) to denote the Lipschitz modulus of h at x ¯, that is, lip (h, x ¯) := lim sup x,z→¯ x

d(h(x), h(z)) . d(x, z)

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Theorem 1. Let X be a complete metric space, Y be a metric space, F : X ⇒ Y be a closed multifunction (i.e., the graph of F is closed) and let (¯ x, y¯) ∈ gph F be given. (i). For any x ∈ / S = F −1 (¯ y ), one has m(x)d(x, S) ≤ φ(x), where the quantity m(x) is defined by ( { m(x) := lim inf |∇|φ(z) :

d(x, z) ≤ (1 − ε)d(x, S), φ(z) φ(x) φ(z) ≤ φ(x), d(z,S) ≤ (1−ε)d(x,S)

ε→0

(11) }) .

(12)

Consequently, if lim inf |∇|φ(x) := m > 0,

x→¯ x, x∈S /

(13)

φ(x) →0 d(x,¯ x)

then there exist reals τ > 0 and δ > 0 such that d(x, S) ≤ τ φ(x)

for all x ∈ B(¯ x, δ).

That is, F is metrically subregular at (¯ x, y¯). (ii). Assume that Y is a Banach space. If lim inf |∇|φ(x) = 0,

x→¯ x, x∈S /

(14)

φ(x) →0 d(x,¯ x)

then there exists a locally Lipschitz mapping h : X → Y with h(¯ x) = 0; lip(h, x ¯) = 0 such that F + h fails to be metrically subregular at (¯ x, y¯). Proof. (i). Given x ∈ / S, since φ is nonnegative on the whole space, then φ(x) ≤ inf φ(u) + φ(x). u∈X

By the Ekeland variational principle, for any ε ∈ (0, 1), take z ∈ X such that d(x, z) ≤ (1 − ε)d(x, S), φ(z) ≤ φ(x); φ(x) φ(z) ≤ φ(u) + (1−ε)d(x,S) d(z, u) ∀u ∈ X. The latter relation implies that φ(z) φ(x) ≤ d(z, S) (1 − ε)d(x, S) and (1 − ε)d(x, S) lim sup u→z

φ(z) − φ(u) ≤ φ(x). d(z, u)

Therefore, one obtains inequality (11), by the definition of m(x). For the second part, suppose that (13) is satisfied. For a given κ ∈ (m−1 , +∞), there exists δ > 0 such that φ(z) |∇|φ(z) > κ−1 for all z ∈ B(¯ x, δ) \ S; < δ. (15) d(z, x ¯) Let x ∈ B(¯ x, δ/2) be given. If φ(x)/d(x, S) ≥ δ, then d(x, S) ≤ φ(x)/δ.

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φ(x) Suppose that φ(x)/d(x, S) < δ. For ε > 0 sufficiently small such as (1−ε)d(x,S) < δ, and for all z ∈ X satisfying φ(z) φ(x) d(z, x) ≤ (1 − ε)d(x, S); ≤ , d(z, S) (1 − ε)d(x, S)

one has z ∈ B(¯ x, δ) \ S, and φ(z) φ(z) φ(x) ≤ ≤ < δ. d(z, x ¯) d(z, S) (1 − ε)d(x, S) By the definition of m(x), one has m(x) ≥ m. Hence, by setting τ := max{κ, δ −1 }, one obtains d(x, S) ≤ τ φ(x) for all

x ∈ B(¯ x, δ/2),

which completes the proof of the first part . (ii). Suppose now (14) is verified. Then, take a sequence {xn } ⊆ X satisfying xn → x ¯; y¯ ∈ / F (xn ); φ(xn )/d(xn , x ¯) < n−2 ; and |∇|φ(xn ) < n−1 /4, ∀n ∈ N. Set tn := d(xn , x ¯), and by picking subsequences if necessary, we can assume that tn+1 < t4n for all tn n. Since |∇|φ(xn ) < n−1 /4, for each n, choose a real ρn with ρn ∈ (0, 3n ) and ρn < nφ(xn )/2 and such that d(x, xn ) , ∀x ∈ B(xn , ρn ). (16) φ(xn ) − φ(x) < 4n For each integer n, by the definition of the lower semicontinuous envelope φ, there exist un ∈ B(xn , ρn /2) and vn ∈ F (un ) such that |∥y¯ − vn ∥ − φ(xn )|
n then d(x, uk ) ≥ d(¯ x, xn ) − d(¯ x, xk ) − d(x, xn ) − d(xk , uk ) > tn − tk − ρn − ρk /2 > tk /2; otherwise, similarly, one also has d(x, uk ) ≥ tk − tn − ρn − ρk /2 > tk /2.

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Consequently,

2t−1 k d(x, uk ) ≥ 1

for all k ̸= n; all x ∈ B(un , ρn /2).

Hence, ( h(x) =

2 − max{1 − 2t−1 n d(x, un ), 0}

ρn (vn − y¯) vn − y¯ − n∥vn − y¯∥

) for all x ∈ B(un , ρn /2).

Moreover, h(¯ x) = 0, by the definition of the sequence {tn }. It implies that d(¯ y , F (un ) + h(un )) ≤ ∥y¯ − vn − h(un )∥ = ρn /n.

(20)

Let us also observe that for any x ∈ B(un , ρn /2), by relations (17) and (18), one has d(¯ y , F (x) + h(x)) ≥ d(¯ y , F (x)) − ∥h(x)∥ ≥ ρn /n − ρn /4n − d(x, xn )/4n ≥ ρn /2n > 0. This shows that x ∈ / (F + h)−1 (¯ y ) for all x ∈ B(un , ρn /2), that is, d(un , (F + h)−1 (¯ y )) ≥ ρn /2. This together with relation (20) shows that F + h is not metrically subregular at (¯ x, y¯). To complete the proof, we demonstrate that h is locally Lipschitz at x ¯ with lip (h, x ¯) = 0. Let ε > 0 be arbitrarily given. Since ∞ ∑

t−1 ¯∥ ≤ n ∥vn − y

n=1

∞ ∑

n−2 < ∞,

n=1

then there exists an index N such that ∞ ∑

t−1 ¯∥ < ε/4. n ∥vn − y

n=N +1 −1 Noticing that the functions 2t−1 n d(·, un ) are Lipschitz with modulus 2tn , respectively, and 2t−1 x, un ) ≥ 1, for all n, take δ > 0 such that for all n = 1, ..., N, all x, y ∈ B(¯ x, δ), one has n d(¯ 2 −1 2 | max{1 − 2t−1 n d(x, un ), 0} − max{1 − 2tn d(y, un ), 0} | ≤

ε d(x, y). 2N

Therefore, we the following estimate holds ∥h(x) − h(y)∥ ≤ ∑N 2 −1 2 | max{1 − 2t−1 ¯∥+ n d(x, un ), 0} − max{1 − 2tn d(y, un ), 0} |∥vn − y ∑n=1 ∞ −1 2 −1 2 | max { 1 − 2t d(x, u ), 0 } − max { 1 − 2t d(y, u ), 0 } |∥ v ¯∥ n n n−y n ∑ n n=N +1 ∞ −1 ≤ εd(x, y)/2 + 2d(x, y) n=N +1 tn ∥vn − y¯∥ < εd(x, y).

As ε is arbitrarily small, we conclude that lip (h, x ¯) = 0. The proof is complete.



Remark 1. The proof of (i) follows the idea from [51, Theorem 2.1]; it is directly based on the Ekeland variational principle (see also, the related works in [4, 29, 49]). The proof of (ii) is inspired from [25, Theorem 3.2-2 ]. In [51], it was established the following sufficient condition (which is x, y¯) : obviously stronger than (13)) for metric subregularity of F at (¯ lim inf |∇|φ(x) := m > 0.

x→¯ x, x∈S / φ(x)→0

Note that this is indeed a sufficient condition for the error bound property of the function φ established in [4, 29].

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3. First order characterizations of the metric subregularity Let X be a Banach space. We use the symbol ∂ to denote any abstract subdifferential, that is any set-valued mapping which associates to every function defined on X and every x ∈ X the set ∂f (x) ⊂ X ⋆ (possibly empty), in such a way that (see [3, 38, 39, 47]) (C1) If f : X → R ∪ {+∞} is a l.s.c convex function, then ∂f coincides with the Fenchel-MoreauRockafellar subdifferential: ∂f (x) := {x∗ ∈ X ∗ : ⟨x∗ , y − x⟩ ≤ f (y) − f (x) ∀y ∈ X }; (C2) ∂f (x) = ∂g(x) if f (y) = g(y) for all y in a neighborhood of x. (C3) Let f : X → R ∪ {+∞} be a l.s.c function and g : X → R be convex and Lipschitz. If f + g attains a local minimum at x0 , then for any ε > 0, there exist x1 , x2 ∈ x0 + εBX , x∗1 ∈ ∂f (x1 ), x∗2 ∈ ∂g(x2 ), such that |f (x1 ) − f (x0 )| < ε and ∥x∗1 + x∗2 ∥ < ε. It is well known that the class of abstract subdifferentials includes Fr´echet subdifferentials in Asplund spaces, viscosity subdifferentials in smooth Banach spaces as well as the Ioffe and the Clarke-Rockafellar subdifferentials in Banach spaces (see [11]). For a closed subset C of X, the normal cone to C with respect to a subdifferential operator ∂ at x ∈ C is defined by N∂ (C, x) = ∂δC (x), where δC is the indicator function of C given by δC (x) = 0 if x ∈ C and δC (x) = +∞ otherwise and we assume here that ∂δC (x) is a cone for any closed subset C of X. Let X, Y be Banach spaces, and let ∂ be a subdifferential on X × Y. Let F : X ⇒ Y be a closed multifunction (graph-closed) and let (¯ x, y¯) ∈ gphF . The multifunction D∗ F (¯ x, y¯) : Y ∗ ⇒ X ∗ defined by D∗ F (¯ x, y¯)(y ∗ ) = {x∗ ∈ X ∗ : (x∗ , −y ∗ ) ∈ N∂ (gph F, (¯ x, y¯))} is called the ∂ −coderivative of F at (¯ x, y¯). Let F : X ⇒ Y be a closed multifunction and as above, for a given (¯ x, y¯) ∈ gph F, denote by φ(x), x ∈ X the lower semicontinuous envelope of the function x 7→ d(¯ y , F (x)), and set m := lim inf |∇|φ(x).

(21)

x→¯ x, x∈S / φ(x) →0 d(x,¯ x)

We next establish a characterization of metric subregularity using abstract coderivatives of multifunctions. In the following theorem, we assume further that ∂ is a subdifferential operator on X × Y which satisfies the separable property in the following sense: (C4) If f (x, y) := f1 (x) + f2 (y), (x, y) ∈ X × Y, where f1 : X → R ∪ {+∞}, f2 : Y → R ∪ {+∞}, is a separable function defined on X × Y, then ∂f (x, y) = ∂f1 (x) × ∂f2 (y),

for all

(x, y) ∈ X × Y.

Theorem 2. Let X, Y be Banach spaces and let ∂ be a subdifferential operator on X × Y. Suppose that F : X ⇒ Y is a closed multifunction between X and Y. For a given (¯ x, y¯) ∈ gph F, if   (¯ x + tu, y¯ + tv) ∈ gph F,        x∗ ∈ D∗ F (¯ x + tu, y¯ + tv)(y ∗ ), ∥y ∗ ∥ = 1,    ∥u∥ = 1, x ¯ + tu ∈ / F −1 (¯ y ), > 0, (22) lim inf sup inf ∥x∗ ∥ :   t↓0+ η>0   t ∥ v ∥ ≤ (1 + η)d(¯ y , F (¯ x + tu)),   v→0     |⟨y ∗ , v ⟩ − ∥v ∥| < η ∥v ∥ then F is metrically subregular at (¯ x, y¯).

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This theorem follows directly from Theorem 1 and the following lemma, which gives an estimate of the quantity m by using the abstract subdifferential operator on X × Y. Lemma 1.

Let ∂ be a subdifferential on X × Y. Then one has   (¯ x + tu, y¯ + tv) ∈ gph F,       ∗ ∗ ∗ ∗   x ∈ D F (¯ x + tu, y ¯ + tv)(y ), ∥ y ∥ = 1,   ∗ −1 ∥u∥ = 1, x ¯ + tu ∈ / F (¯ y ), m ≥ lim inf sup inf ∥x ∥ : .   t↓0+ η>0   t ∥ v ∥ ≤ (1 + η)d(¯ y , F (¯ x + tu)),   v→0     |⟨y ∗ , v ⟩ − ∥v ∥| < η ∥v ∥

(23)

Proof. Let {xn } ⊆ X be a sequence such that xn → x ¯ , xn ∈ / F −1 (¯ y ), lim

n→∞

φ(xn ) = 0, and lim |∇|φ(xn ) := m. n→∞ ∥xn − x ¯∥

Take a sequence of nonnegative reals {εn } such that εn ∈ (0, φ(xn )) and εn /φ(xn ) → 0. Then for each n, there is ηn ∈ (0, εn ) with 2ηn + εn < φ(xn ) and 1 − (m + εn + 2)ηn > 0 such that d(¯ y , F (z)) ≥ φ(xn )(1 − εn ), ∀z ∈ B(xn , 4ηn ) and m + εn ≥

φ(xn ) − φ(z) ∥xn − z ∥

¯ n , ηn ). for all z ∈ B(x

Equivalently, φ(xn ) ≤ φ(z) + (m + εn )∥z − xn ∥

¯ n , ηn ). for all z ∈ B(x

Take zn ∈ B(xn , ηn2 /4), wn ∈ F (zn ) such that ∥y¯ − wn ∥ ≤ φ(xn ) + ηn2 /4. Then, ¯ n , ηn ). ∥y¯ − wn ∥ ≤ φ(z) + (m + εn )∥z − xn ∥ + ηn2 /4 ∀z ∈ B(x Therefore, ∥y¯ − wn ∥ ≤ ∥y¯ − w∥ + δgphF (z, w) + (m + εn )∥z − zn ∥ + (m + εn + 1)ηn2 /4

¯ n , ηn ) × Y. ∀(z, w) ∈ B(x

Applying the Ekeland variational principle to the function (z, w) 7→ ∥y¯ − w∥ + δgphF (z, w) + (m + ε)∥z − zn ∥ ¯ n , ηn ) × Y, we can select (zn1 , wn1 ) ∈ (zn , wn ) + ηn BX×Y with (zn1 , wn1 ) ∈ gphF such that on B(x 4 ∥y¯ − wn1 ∥ ≤ ∥y¯ − wn ∥(≤ φ(xn ) + η 2 /4);

(24)

and that the function (z, w) 7→ ∥y¯ − w∥ + δgphF (z, w) + (m + εn )∥z − zn ∥ + (m + εn + 1)ηn ∥(z, w) − (zn1 , wn1 )∥ ¯ n , ηn ) × Y at (zn1 , wn1 ). Hence, by (C3), choose attains a minimum on B(x wn2 ∈ BY (wn1 , ηn ); (zn3 , wn3 ) ∈ BX×Y ((zn1 , wn1 ), ηn ) ∩ gphF ; wn2∗ ∈ ∂ ∥y¯ − ·∥(wn2 ); (zn3∗ , −wn3∗ ) ∈ N (gphF, (zn3 , wn3 )) satisfying ∥wn2∗ − wn3∗ ∥ < (m + εn + 2)ηn

and ∥zn3∗ ∥ ≤ m + εn + (m + εn + 2)ηn .

(25)

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Since wn2∗ ∈ ∂ ∥y¯ − ·∥(wn2 ) (note that ∥y¯ − wn2 ∥ ≥ ∥y¯ − wn ∥ − ∥wn2 − wn ∥ ≥ φ(xn ) − εn − 2ηn > 0), then ∥wn2∗ ∥ = 1 and ⟨wn2∗ , wn2 − y¯⟩ = ∥y¯ − wn2 ∥. Thus, from the first relation in (25), it follows that ∥wn3∗ ∥ ≥ ∥wn2∗ ∥ − (m + εn + 2)ηn = 1 − (m + εn + 2)ηn ; ∥wn3∗ ∥ ≤ ∥wn2∗ ∥ + (m + εn + 2)ηn = 1 + (m + εn + 2)ηn .

Set tn = ∥zn3 − x ¯∥; un = (zn3 − x ¯)/tn ; vn = (wn3 − y¯)/tn , and

yn∗ = wn3∗ /∥wn3∗ ∥; x∗n = zn3∗ /∥wn3∗ ∥.

Since φ(xn )(1 − εn ) ≤ d(¯ y , F (¯ x + tn un )) ≤ tn ∥vn ∥ ≤ ∥y¯ − wn1 ∥ + ηn ≤ φ(xn ) + ηn2 + 5ηn /4; and tn = ∥zn3 − x ¯∥ ≥ ∥xn − x ¯∥ − ∥zn3 − xn ∥ ≥ ∥xn − x ¯∥ − ηn2 /4 − 5ηn /4 > 0, then ∥vn ∥ ≤

φ(xn ) + ηn2 /4 + ηn . ∥xn − x ¯∥ − ηn2 /4 − 5ηn /4

As φ(xn )/∥xn − x ¯∥ → 0 as well as ηn /φ(xn ) → 0, one obtains lim vn = 0.

n→∞

(26)

Next one has x∗n ∈ D∗ F (¯ x + tn un , y¯ + tn vn )(yn∗ ) with ∥yn∗ ∥ = 1 and by the second relation of (25), one derives that m + εn + (m + εn + 2)ηn ∥x∗n ∥ = ∥zn3∗ ∥/∥wn3∗ ∥ ≤ . (27) 1 − (m + εn + 2)ηn On the other hand, tn ⟨yn∗ , vn ⟩ = ≥

2∗ 2 2∗ 3 2 3∗ 2∗ 3 ⟨wn ,wn −¯ y ⟩+⟨wn ,wn −wn ⟩+⟨wn −wn ,wn −¯ y⟩ 3∗ ∥ ∥wn 2 3 ∥¯ y −wn ∥−2ηn −(m+εn +2)ηn ∥¯ y −wn ∥ n +2)ηn )tn ∥vn ∥−4ηn ≥ (1−(m+ε . 1+(m+εn +2)ηn 1+(m+εn +2)ηn

Hence, |⟨yn∗ , vn ⟩ − ∥vn ∥| ≤

2(m + εn + 2)ηn ∥vn ∥ + 4ηn . (1 + (m + εn + 2)ηn )tn

(28)

Since εn /φ(xn ) → 0 and ηn ∈ (0, εn ), and ηn ηn ≤ → 0 as n → ∞, tn ∥vn ∥ φ(xn )(1 − εn ) by combining relations (26), (27), (28), we complete the proof.



From Theorem 2, one obtains the following sufficient condition for the error bound of a lower semicontinuous function f : X → R ∪ {+∞}. Let ∂ be an abstract subdifferential on X × R. As usual, epi f denotes the epigraph of f. In the literature, the subdifferential with respect to ∂ of a lower semicontinuous function f : X → R ∪ {+∞} at x ∈ Dom f is denoted and defined by ∂f (x) := {x∗ : (x∗ , −1) ∈ N∂ (epi f, (x, α)), f (x) ≤ α}. If x ∈ / Dom f, we set ∂f (¯ x) = ∅.

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Corollary 1. Let f : X → R ∪ {+∞} be a lower semicontinuous function. Let x ¯ ∈ S := {x ∈ X : f (x) ≤ 0} be given. If lim inf ∂f (x) > 0, x→¯ x, x∈S /

f (x)/∥x−¯ x∥→0

then there exist τ, δ > 0 such that d(x, S) ≤ τ [f (x)]+

for all x ∈ B(¯ x, δ).

(29)

Where, [f (x]+ := max{f (x), 0}. Proof. Let F : X ⇒ R be a multifunction defined by gph F := epi f. Obviously, the metric subreg¯. By noticing that for ularity of f at (¯ x, 0) is equivalent to the error bound property (29) of f at x (x, α) ∈ epi f, D∗ F (x, α)(1) ⊆ ∂f (x), the conclusion follows immediately from Theorem 2



It is worth to note that Corollary 1 covers some results on the error bound property in the current literature (see [4, 25, 58]). Let us recall from [25] the notion of limit set critical for metric subregularity of a set-valued mapping. The definition in [25] is stated for the Frech´et subdifferential, however it is also valid for any subdifferential operator. Definition 1. Let X, Y be Banach spaces and let ∂ be a subdifferential operator on X × Y. Let F : X ⇒ Y be a closed multifunction. For a given (¯ x, y¯) ∈ gph F, the limit set critical for metric subregularity of F at (¯ x, y¯) denoted by CrF (¯ x, y¯) is defined as the set of all (v, x∗ ) ∈ Y × X ∗ such that there exist sequences {tn } ↓ 0, (vn , x∗n ) → (v, x∗ ), (un , yn∗ ) ∈ SX × SY ∗ with x∗n ∈ D∗ F (¯ x+ ∗ ∗ ∗ tn un , y¯ + tn vn )(yn ), where, SX , SY stand for the unit spheres in X, Y , respectively. In [25], the author has established that (0, 0) ∈ / CrF (¯ x, y¯) is a sufficient condition for metric subregularity of F at (¯ x, y¯). Theorem 2 permits us to obtain a sufficient condition which is weaker the one mentioned in [25] . We present a sharper version of the notion above in the following definition. Definition 2. For a closed multifunction F : X ⇒ Y and (¯ x, y¯) ∈ gph F, the strict limit set critical for metric subregularity of F at (¯ x, y¯) denoted by SCrF (¯ x, y¯) is defined as the set of all (v, x∗ ) ∈ Y × X ∗ such that there exist sequences {tn } ↓ 0, (vn , x∗n ) → (v, x∗ ), (un , yn∗ ) ∈ SX × SY ∗ ∗ n ,vn ⟩ with x∗n ∈ D∗ F (¯ x + tn un , y¯ + tn vn )(yn∗ ), y¯ ∈ / F (¯ x + tn un ) (∀n), and ⟨y∥v → 1. n∥ Obviously, SCrF (¯ x, y¯) ⊆ CrF (¯ x, y¯). Theorem 2 yields immediately the following corollary. Corollary 2. Let X, Y be Banach spaces and let ∂ be a subdifferential operator on X × Y. Let F : X ⇒ Y be a closed multifunction between X and Y and let (¯ x, y¯) ∈ gph F. If (0, 0) ∈ / SCrF (¯ x, y¯) then F is metrically subregular at (¯ x, y¯). Example 1.

Let the mapping F : R → R2 defined by ) {( x sin x1 , x3 sin x1 if x ̸= 0 F (x) = 0 otherwise

For this mapping, we show that (0, 0) ∈ / SCrF (0, 0), therefore F is metrically subregular at (0, 0); however, (0, 0) ∈ CrF (0, 0).

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Indeed, the mapping F is continuously differentiable at all x ̸= 0 and for y ∗ := (α, β) ∈ R2 , one has ( ) ( ) 1 1 1 1 1 ∗ ∗ ′ ∗ 2 D F (x)(y ) = ⟨F (x), y ⟩ = α sin − cos + β 3x sin − x cos . x x x x x Let sequences (tn ) → 0+ ; (un ) ⊆ R with |un | = 1. By setting xn := tn un , then ( ) 1 1 vn = un sin , x2n sin . xn xn Let (yn∗ ) := (αn , βn ) ⊆ R2 with |αn | + |βn | = 1 be a sequence such that 1 1 2 ⟨yn∗ , vn ⟩ αn sin xn + βn xn sin xn = → 1. ∥vn ∥ | sin x1n | + |x2n sin x1n |

Then, αn → 1 and βn → 0. It follows that, if vn → 0 then sin x1n → 0, and therefore, ∗ |x∗n | = | D∗ F ) ( ) ( (xn )(yn )| = αn sin x1n − x1n cos x1n + βn 3x2n sin x1n − xn cos x1n → +∞,

which provides (0, 0) ∈ / SCrF (0, 0). To show (0, 0) ∈ CrF (0, 0), let us take a sequence of positives (εn ) → 0+ . By setting tn = xn :=

1 ; 2nπ − εn

yn∗ = (0, 1)(∀n),

then, F (xn ) ̸= 0 (when n is sufficiently large) and we obtain ( ) 1 1 1 1 vn = sin , x3n sin → 0 as well as x∗n = 3x2n sin − xn cos → 0, xn xn xn xn which implies (0, 0) ∈ CrF (0, 0). Let us now consider the mixed smooth- convex inclusion of the form: 0 ∈ g(x) − F (x) := G(x),

(30)

where g : X → Y is a mapping of C 1 class, i.e., the class of continuously differentiable mappings, around x ¯ ∈ G−1 (0); F : X ⇒ Y is a closed convex multifunction. By the standard sum rule for the ¯, one has coderivative (see, e.g., [44]), for (x, y) ∈ gph G with x near x D∗ G(x, y)(y ∗ ) = Dg(x)∗ (y ∗ ) + D∗ F (x, g(x) − y)(−y ∗ ), y ∗ ∈ Y ∗ .

(31)

In this case, the condition (0, 0) ∈ / SCrG(¯ x, 0) is also a necessary condition for the metric subregularity, as showed in the following proposition. Proposition 1. With the assumptions as above, the multifunction G := g − F is metrically subregular at (¯ x, 0) if and only if (0, 0) ∈ / SCrG(¯ x, 0). Proof. It suffices to prove the necessary part. Suppose that G is metrically subregular at (¯ x, 0). There are τ > 0, δ > 0 such that d(x, G−1 (0)) ≤ τ d(0, G(x)) ∀x ∈ B(¯ x, δ).

(32)

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Take ε ∈ (0, τ −1 ). Since g is of C 1 class around x ¯, there is, say the same δ as above, such that ∥g(x1 ) − g(x2 ) − Dg(x1 )(x1 − x2 )∥ ≤ ε∥x1 − x2 ∥ ∀x1 , x2 ∈ B(¯ x, δ).

(33)

Let sequences (tn ), (un ), (vn ), (x∗n ), (yn∗ ) such that (tn ) ↓ 0; (un , yn∗ ) ∈ SX × SY ∗ ; x∗n ∈ D∗ G(¯ x+ ∗ ⟨yn ,vn ⟩ ∗ ∗ tn un , tn vn )(yn ); 0 ∈ / G(¯ x + tn un ) (∀n); (vn ) → 0 and ∥vn ∥ → 1. We will prove that (xn ) does not converge to 0. Indeed, pick a sequence (εn ) ↓ 0, by assuming without loss of generality τ (1 + εn )tn ∥vn ∥ ≤ tn < δ/2, for each n, there exists zn ∈ G−1 (0) such that ∥zn − x ¯ − tn un ∥ ≤ τ (1 + εn )d(0, G(¯ x + tn un )) ≤ τ (1 + εn )tn ∥vn ∥.

(34)

Consequently, zn ∈ B(¯ x, δ), for all n. According to (31), one has x∗n = Dg(¯ x + tn un )∗ (yn∗ ) + zn∗ , with zn∗ ∈ D∗ F (¯ x + tn un , g(¯ x + tn un ) − tn vn )(−yn∗ ). Since F is a convex multifunction, then ⟨zn∗ , z − x ¯ − tn un ⟩ + ⟨yn∗ , w − g(¯ x + tn un ) + tn vn ⟩ ≤ 0 ∀(z, w) ∈ gph F.

By taking (z, w) := (zn , g(zn )) into account, one has ⟨x∗n , x ¯ + tn un − zn ⟩ ≥ −⟨yn∗ , g(¯ x + tn un ) − g(zn ) − Dg(¯ x + tn un )(¯ x + tn un − zn )⟩ + tn ⟨yn∗ , vn ⟩.

Therefore, from relations (33), (34), one obtains τ (1 + εn )tn ∥vn ∥∥x∗n ∥ ≥ ⟨x∗n , zn − x ¯ − tn un ⟩ ≥ tn ⟨yn∗ , vn ⟩ − ετ (1 + εn )tn ∥vn ∥. This implies that lim inf n→∞ ∥x∗n ∥ ≥ (1 − τ ε) > 0, which ends the proof.



Remark 2. (i) As the condition (0, 0) ∈ / SCrG(¯ x, 0), implies condition (22) in Theorem 2, which implies condition (13) in Theorem 1, this proposition also tells us that (13) in Theorem 1 as well as condition (22) in Theorem 2 are necessary and sufficient conditions for metric subregularity of the smooth-convex inclusion (30). (ii) When F is a constant mapping, i.e., F (x) = C for all x ∈ X, where C is a nonempty closed convex subset of Y, it was well-known that the metric regularity of G := g − C around (¯ x, 0) ∈ gph G is equivalent to the Robinson constraint qualification 0 ∈ int [Dg(¯ x)X − (C − g(¯ x))], which is equivalent to inf

∗ y ∗ ∈N (C,g(¯ x))∩SY

(35)

∥Dg(¯ x)∗ y ∗ ∥ > 0,

where, N (C, g(¯ x)) stands for the normal cone to C at g(¯ x). So in this case, the Robinson constraint qualification implies the condition (0, 0) ∈ / SCrG(¯ x, 0). Especially, for constraint systems of smooth inequalities and equalities (2), the Robinson constraint qualification is equivalent to the Mangasarian-Fromovitz constraint qualification (MFCQ). For this generalized equation (2): 0 ∈ F (x) given by smooth inequalities and equalities, by a straightforward calculation, we derive that the condition (0, 0) ∈ / SCrF (¯ x, 0) is equivalent to the following    ∑  ∑ lim inf inf ∥ ∇fi (x)∥ : |λi | = 1, λi fi (x) ≥ 0, i ∈ I(x) > 0.   x→¯ x,x∈F / −1 (0) i∈I(x)

i∈I(x)

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Where, I(x) := {i ∈ {1, ..., k + d} : |fi (x)| = f (x), i ∈ {1, ..., k }; fi (x) = f (x), i ∈ {k + 1, ..., k + d}}; f (x) := max{|f1 (x)|, ..., |fk (x)|, fk+1 (x), ..., fk+d (x)}. This latter condition is indeed the well-known necessary and sufficient condition for the error bound property of the system (2) described by smooth functions (see, for example [4, 50, 59]). For the smooth-convex inclusion, as showed in the following proposition, which generalizes Proposition 3.9 in [25], the condition (0, 0) ∈ / CrG(¯ x, 0) is also actually a sufficient and necessary condition for either the strong metric subregularity or the metric regularity of G at (¯ x, 0). Proposition 2. For the mixed smooth-convex inclusion (30) and for given x ¯ ∈ G−1 (0), (0, 0) ∈ / CrG(¯ x, 0) if and only if G is either strongly metrically subregular or metrically regular at (¯ x, 0). Proof. The necessary part was proved in ([25], Proposition 3.8), it is valid for any closed multifunction G. For the sufficient part, suppose that (0, 0) ∈ / CrG(¯ x, 0), and assume that G is not strongly metrically subregular at (¯ x, 0). Then there exists a sequence (zn ) → x ¯ with zn ∈ G−1 (0), zn ̸= x ¯. Assume to contrary that G is not metrically regular at (¯ x, 0). By the coderivative characterization of metric regularity (see, e.g., [5]), take sequences (xn , yn ) → (¯ x, 0) with (xn , yn ) ∈ gph G; ∗ ∗ ∗ ∗ ∗ xn ∈ D G(xn , yn )(yn ) with ∥yn ∥ = 1 such that ∥xn ∥ → 0. Then, there exists zn∗ ∈ D∗ F (xn , g(xn ) − yn )(−yn∗ ) such that x∗n = Dg(xn )∗ (yn∗ ) + zn∗ . By taking a subsequences if necessary, we can assume that zn ̸= xn for all n, and that ∥x∗n ∥ < 1/(3n2 ); ∥yn ∥ < rn /(3n2 ); ∥g(zn ) − g(xn ) − Dg(¯ x)(zn − xn )∥ < rn /(3n2 ),

where, rn := ∥zn − xn ∥ > 0. Therefore, ⟨zn∗ , zn ⟩ + ⟨yn∗ , g(zn )⟩ = ⟨zn∗ , xn ⟩ + ⟨yn∗ , g(xn ) − yn ⟩ +⟨x∗n , zn − xn ⟩ + ⟨yn∗ , g(zn ) − g(xn ) − Dg(xn )(zn − xn )⟩ + ⟨yn∗ , yn ⟩ . > ⟨zn∗ , xn ⟩ + ⟨yn∗ , g(xn ) − yn ⟩ − rn /n2 .

Since zn∗ ∈ D∗ F (xn , g(xn ) − yn )(−yn∗ ), then ⟨zn∗ , z − xn ⟩ + ⟨yn∗ , w − g(xn ) + yn ⟩ ≤ 0 ∀(z, w) ∈ gph F.

(36)

The later relations imply that ⟨zn∗ , −zn ⟩ + ⟨yn∗ , −g(zn )⟩
0, there is a neighborhood U of the origin such that (S − x ¯) ∩ U ⊂ [T (S; x ¯)]ε where [T (S; x ¯)]ε := {x ∈ X : d(x/∥x∥, T (S; x ¯)) < ε} ∪ {0} is the ε-conic neighborhood of T (S; x ¯). It should note that in a finite dimensional space, every nonempty set is tangentiable at any point ( see [20]). Lemma 2. Let S ⊂ X be nonempty, x ¯ ∈ S and {xn } ⊂ S \ {x ¯}. Assume that S is tangentiable } { xn −¯ x at x ¯, T (S, x ¯) is locally compact at the origin and {xn } converges to x ¯. Then the sequence ∥xn −¯x∥ has a convergent subsequence. Proof. Since S is tangentiable at x ¯, by passing to a subsequence if necessary, we may assume that ( ) xn − x ¯ 1 d , T (S; x ¯) < , ∀n. ∥xn − x ¯∥ n Then for every n, there exists un ∈ T (S, x ¯) such that

xn − x

1 ¯

< . − u n

∥xn − x

n ¯∥ Since T (S, x ¯) is a cone and locally { compact } at 0, there exists a subsequence {ukn } ⊂ {un } which x −¯ x converges to some point u. Then ∥xkn −¯x∥ is also converges to u.  kn

We note that in an infinite dimensional space there exist first- order tangentiable sets with their tangent cones are locally compact at the origin such that they are not contained in any finite dimensional subspace. For instant let H be an infinite dimensional Hilbert space with a countable base {e1 , e2 , . . . , en , . . .} such that { 1, i = j; ⟨ei , ej ⟩ = 0, i ̸= j.

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Denote S := {en : n = 2, 3, ...} ∪ {te1 : t ≥ 0}. Then T (S; 0) = {te1 : t ≥ 0}. One can see that S is first- order tangentiable at 0, T (S; 0) is locally compact at 0 and no finite dimensional subspace of H contains S. Now let G : X ⇒ Y be a multifunction. The following two propositions, giving metric subregularity of the contingent derivative of a metrically regular multifunction, are generalizations of Proposition 2.1 in [25]. The first is for a general multifunctions. Here, instead of supposing that X is finite dimensional, we assume that G−1 (¯ y ) is tangentiable at x ¯ and T (G−1 (¯ y ), x ¯) is locally compact at the origin. The closed unit ball in X is denoted by BX . Proposition 3. Let G : X ⇒ Y be a set-valued map from X to another normed space Y . Assume that G is metrically subregular at (¯ x, y¯) ∈ gph G with some modulus κ. If G−1 (¯ y ) is tangen−1 tiable at x ¯ and T (G (¯ y ), x ¯) is locally compact at the origin then the contingent derivative CG(¯ x, y¯) is metrically subregular at (0, 0) with modulus κ. Proof. The proof of this proposition is direct from definition, which is similar to the one of Proposition 2.1 in [25]. Let u ∈ X and ϵ > 0 be arbitrary. Choose v ∈ CG(¯ x, y¯)(u) such that ∥v ∥ < d(0, CG(¯ x, y¯)(u)) + ϵ. Since (u, v) ∈ T (gph G, (¯ x, y¯)) there are sequences tn ↓ 0 and (un , vn ) → (u, v) such that (¯ x + tn un , y¯ + tn vn ) ∈ gph G. We have d(¯ x + tn un , G−1 (¯ y )) ≤ κd(¯ y , G(¯ x + tn un )) ≤ κtn ∥vn ∥ ≤ κtn [d(0, CG(¯ x, y¯)(u)) + ϵ] for k sufficiently large. Then choose xn ∈ G−1 (¯ y ) such that xn − x ¯ − tn un ∈ κtn [d(0, CG(¯ x, y¯)(u)) + ∥xn −¯ x∥ 2ϵ]BX . This implies the boundedness of the sequence { tn }. Hence we may assume that { ∥xntn−¯x∥ } converges to some α. On the other hand, by Lemma 2, we also may assume that the sequence x { ∥xxnn −¯ } converges to some point a. Set u ¯n := xntn−¯x . Then u ¯n ∈ un + κ[d(0, CG(¯ x, y¯)(u)) + 2ϵ]BX −¯ x∥ and u ¯n → u ¯ := αa. Therefore u ¯ ∈ u + κ[d(0, CG(¯ x, y¯)(u)) + 2ϵ]BX . Since y¯ ∈ G(¯ x + tn u ¯n ) we have 0 ∈ CG(¯ x, y¯)(¯ u). Thus d(u, CG(¯ x, y¯)−1 (0)) ≤ ∥u − u ¯∥ ≤ κ[d(0, CG(¯ x, y¯)(u)) + 2ϵ]. Take the limit on ϵ and the proof is complete.



Now consider again the following mixed constraint system: 0 ∈ g(x) − F (x),

(37)

where, as the preceding section, F : X ⇒ Y is a closed and convex set-valued map and g : X → Y is assumed to be continuously differentiable in a neighbourhood of a point x ¯ ∈ (g − F )−1 (0). Set G(x) := g(x) − F (x) and C := CG(¯ x, 0)−1 (0) = {u ∈ X : Dg(¯ x)(u) ∈ CF (¯ x, g(¯ x))(u)}.

(38)

Proposition 4. For the mixed smooth-convex constraint system (37), and for a given x ¯∈ −1 G (0) := (g − F ) (0), if G is metrically subregular at (¯ x, 0) and if we assume that X is reflexive, then CG(¯ x, 0) is metrically subregular at (0, 0) with the same modulus as G. −1

Proof. Suppose that G is metrically subregular at (¯ x, 0) with modulus κ. Firstly, note that for the mixed smooth-convex constraint system (37), one has CG(¯ x, 0)(u) = Dg(¯ x)(u) − CF (¯ x, g(¯ x))(u), u ∈ X.

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As in the proof of Proposition 3, for given u ∈ X, ε > 0, take v ∈ CG(¯ x, 0)(u) such that ∥v ∥ < d(0, CG(¯ x, 0)(u)) + ϵ, and choose sequences tn ↓ 0 and (un , vn ) → (u, v) such that (¯ x + tn un , tn vn ) ∈ gph G. For each n, there exists then xn ∈ G−1 (0) such that ∥xn − x ¯ − tn un ∥ ≤ κtn [d(0, CG(¯ x, 0)(u)) + 2ϵ].

By setting u ¯n :=

xn −¯ x , tn

since {u ¯n } is bounded and X is reflexive, then by passing to { a subsequence } if

necessary, we may assume that {u ¯n } weakly converges to some u ¯ ∈ X. Therefore, also weakly converges to Dg(¯ x)(¯ u). Since gph F is convex, then

g(¯ x+tn u ¯n )−g(¯ x) tn

(¯ u, Dg(¯ x)(¯ u)) ∈ clw cone(gph F − (¯ x, g(¯ x))) = cl cone(gph F − (¯ x, g(¯ x))) = T (gph F, (¯ x, g(¯ x))), where, clw A, clA, and coneA stand respectively for the weak closure, the closure and the cone hull of some subset A. Consequently, u ¯ ∈ CG(¯ x, 0)−1 (0), and one has d(u, CG(¯ x, 0)−1 (0)) ≤ ∥u − u ¯∥ ≤ κ[d(0, CG(¯ x, 0)(u)) + 2ϵ]. 

As ε > 0 is arbitrary, this completes the proof. In what follows, as in ([25]), we make use of the following two assumptions.

Assumption 1. There exist η, R > 0 such that for every x, x′ ∈ B(¯ x, R), the following inequality holds ∥g(x) − g(x′ ) − Dg(¯ x)(x − x′ )∥ ≤ η max{∥x − x ¯∥, ∥x′ − x ¯∥}∥x − x′ ∥.

Assumption 2. The second order directional derivative g ′′ (¯ x; u) := lim

t→0+

g(¯ x + tu) − g(¯ x) − tDg(¯ x)(u) 2 t /2

exists for every u ∈ C and the convergence is uniform with respect to u in bounded subsets of C . Obviously, assumptions 1 and 2 yield the following relation ∥g ′′ (¯ x, u)∥ ≤ 2η, ∀u ∈ C ∩ SX .

(39)

Let us recall from ([25]) the notion of inner second order approximation mappings for convex sets. Definition 3. ([25]) Let S be a closed convex subset of a Banach space Z, A : X → Z be a continuous linear map and s ∈ S, u ∈ A−1 (T (S; s)). Let ξ be a nonnegative real number. A set I ⊂ Z is called an inner second order approximation set for S at s with respect to A, u and ξ if lim t−2 d(s + tAu +

t→0+

t2 w, S + t2 ξABX ) = 0 2

(40)

holds for all w ∈ I . A set-valued map AS,s,A,ξ : A−1 (T (S; s)) ∩ SX ⇒ Z is an inner second order approximation mapping for S at s with respect to A, ξ if for each u ∈ A−1 (T (S; s)) ∩ SX the set AS,s,A,ξ (u) is an inner second order approximation set with respect to A, u and ξ and the limit (40) holds uniformly for all u ∈ A−1 (T (S; s)) ∩ SX and all w ∈ AS,s,A,ξ (u).

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Denote by IX the identify map of X. It can be seen that C = (IX , Dg(¯ x))−1 (T (gph F, (¯ x, g(¯ x)))).

As usual, the support function of a set C ⊆ X is denoted by σC : X ∗ → R ∪ {+∞}, and is defined by σC (x∗ ) := sup⟨x∗ , x⟩, x∗ ∈ X ∗ . x∈C

The norm in the space X × Y is defined by ∥(x, y)∥ := ∥x∥ + ∥y ∥. Theorem 3. Suppose that X, Y are Banach spaces and Assumptions 1 and 2 are fulfilled. 1. If the contingent derivative CG(¯ x, 0) is metrically subregular at (0, 0) and there are real ξ ≥ 0 and an inner second order approximation map A for gph F at (¯ x, g(¯ x)) with respect to (IX , Dg(¯ x)) and ξ such that for each sequence {(xn ∗ , yn ∗ )} ⊂ X ∗ × SY ∗ satisfying lim [⟨(xn ∗ , yn ∗ ), (¯ x, g(¯ x))⟩ − σgph F (xn ∗ , yn∗ )] = lim ∥Dg(¯ x)∗ yn ∗ + xn ∗ ∥ = 0

n→∞

one has

n→∞

lim inf sup {⟨yn ∗ , g ′′ (¯ x, u)⟩ − σA(u) (xn ∗ , yn ∗ )} < 0, n→∞ u∈C∩S X

(41)

then G is metrically subregular at (¯ x, 0). 2. Conversely, if G is metrically subregular at (¯ x, 0) and lim sup t→0+

d((¯ x + tu, g(¯ x) + tDg(¯ x)(u)), gph F ) t2

(42)

is bounded on C ∩ SX and the convergence is uniform for all u ∈ C ∩ SX , then there are real ξ ≥ 0 and an inner second order approximation map A for gph F at (¯ x, g(¯ x)) with respect to (IX , Dg(¯ x)) and ξ such that for each sequence {(xn ∗ , yn ∗ )} ⊂ X ∗ × SY ∗ satisfying lim [⟨(xn ∗ , yn ∗ ), (¯ x, g(¯ x))⟩ − σgph F (xn ∗ , yn ∗ )] = lim ∥Dg(¯ x)∗ yn ∗ + xn ∗ ∥ = 0,

n→∞

one has

n→∞

x, u)⟩ − σA(u) (xn ∗ , yn ∗ )} ≤ 0. lim inf sup {⟨yn ∗ , g ′′ (¯ n→∞ u∈C∩S X

(43)

Moreover, if G−1 (0) is tangentiable at x ¯ and the tangent cone T (G−1 (0), x ¯) is locally compact at the origin then the contingent derivative CG(¯ x, 0) is metrically subregular at (0, 0). To prove this theorem, we need the following lemma. Lemma 3. Suppose that Assumption 1, 2 are fulfilled and G is metrically subregular at (¯ x, 0). If (42) is bounded on C ∩ SX and the convergence is uniform for all u ∈ C ∩ SX , then the mapping A(u) := {(0, g ′′ (¯ x, u))}, u ∈ C ∩ SX , is an inner second order approximation mapping for gph F at (¯ x, g(¯ x)) with respect to (IX , Dg(¯ x)) and some ξ > 0. Proof. Let a sequence tn → 0+ and u ∈ C ∩ SX be given. Since (42) is bounded on C ∩ SX and the convergence is uniform, there is a sequence (¯ x + tn u ¯n , g(¯ x) + tn vn ) ∈ gph F with ∥(tn (¯ un − u), tn (vn − Dg(¯ x)(u)))∥ ≤ γtn 2

(44)

for some constant γ. By the metric subregularity of G, there is un ∈ X with x ¯ + tn un ∈ G−1 (0) and there is κ > 0 such that, for n sufficiently large, d(¯ x + tn u ¯n , x ¯ + tn un ) ≤ κd(0, G(¯ x + tn u ¯n )) + tn 2 .

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18

Then taking (44) and Assumption 1 into account, one has tn ∥ u ¯n − un ∥ ≤ κd(g(¯ x + tn u ¯n ), F (¯ x + tn u ¯n )) + tn 2 ≤ κd(g(¯ x + tn u ¯n ), g(¯ x) + tn vn ) + tn 2 = κ∥g(¯ x + tn u ¯n ) − g(¯ x) − tn vn ∥ + tn 2 ≤ κ∥g(¯ x + tn u ¯n ) − g(¯ x) − tn Dg(¯ x)(¯ un )∥ + κ∥tn (Dg(¯ x)(¯ un ) − vn )∥ + tn 2 2 2 ≤ κηtn ∥u ¯n ∥ + κ∥tn Dg(¯ x)(¯ un − u)∥ + κ∥tn (Dg(¯ x)(u) − vn )∥ + tn 2 2 2 2 2 ≤ 2κηtn + κγ ∥Dg(¯ x)∥tn + κγtn + tn which together with (44) implies ∥un − u∥ ≤ ξtn

(45)

for some constant ξ. Next, by using Assumption 1, we have the following estimation tn 2 (0, g ′′ (¯ x, u)), gph F + tn 2 ξ(IX , Dg(¯ x))BX ) 2 tn 2 2 tn ′′ ≤ 2 d((¯ x + tn u, g(¯ x) + tn Dg(¯ x)(u) + g (¯ x, u)), gph F − tn (IX , Dg(¯ x))(un − u)) 2 tn 2 2 tn ′′ = 2 d((¯ x + tn un , g(¯ x) + tn Dg(¯ x)(un ) + g (¯ x, u)), gph F ) 2 tn tn 2 ′′ 2 x + tn un , g(¯ x) + tn Dg(¯ x)(un ) + ≤ 2 d((¯ g (¯ x, u)), {x ¯ + tn un } × F (¯ x + tn un )) 2 tn 2 tn ′′ 2 x) + tn Dg(¯ x)(un ) + g (¯ x, u)), F (¯ x + tn un )) = 2 d(g(¯ 2 tn 2 tn ′′ 2 x + tn un ) − g(¯ x) − tn Dg(¯ x)(un ) − ≤ 2 ∥g(¯ g (¯ x, u)∥ 2 tn 2 x + tn un ) − g(¯ x + tn u) − tn Dg(¯ x)(un − u)∥+ ≤ 2 ∥g(¯ tn 2 tn 2 ′′ + 2 ∥g(¯ x + tn u) − g(¯ x) − tn Dg(¯ x)(u) − g (¯ x, u)∥ 2 tn 2 x + tn u) − g(¯ x) − tn Dg(¯ x)(u)] − g ′′ (¯ x, u)∥. ≤ 2η ∥un ∥∥un − u∥ + ∥ 2 [g(¯ tn 2

x, g(¯ x)) + tn (u, Dg(¯ x)(u)) + 2 d((¯

By (45) and Assumption 2 the last right hand part of the above inequalities converges to 0 as n → ∞ uniformly for all u ∈ C ∩ SX . Therefore A(u) := {(0, g ′′ (¯ x, u))}, ∀u ∈ C ∩ SX is an inner second order approximation map for gph F at (¯ x, g(¯ x)) with respect to (IX , Dg(¯ x)) and ξ .  Proof of Theorem 3. 1. Suppose on the contrary that G is not metrically subregular at (¯ x, 0). Then by Theorem 2, there exist sequences xn → x ¯, ϵn → 0, yn ∈ F (xn ), yn∗ ∈ SY ∗ , x∗n ∈ D∗ F (xn , yn )(−yn∗ ) such that d(0, g(xn ) − F (xn )) g(xn ) ∈ / F (xn ), → 0, ∥Dg(¯ x)∗ yn∗ + x∗n ∥ → 0, (46) ∥xn − x ¯∥ ∥g(xn ) − yn ∥ ≤ (1 + ϵn )d(0, g(xn ) − F (xn )), (47) |⟨yn∗ , g(xn ) − yn ⟩ − ∥g(xn ) − yn ∥| ≤ ϵn ∥g(xn ) − yn ∥.

(48)

Immediately, from definitions of x∗n , yn∗ and from (46), (47), we have lim [⟨(x∗n , yn∗ ), (¯ x, g(¯ x))⟩ − σgph F (x∗n , yn∗ )] = lim ⟨(x∗n , yn∗ ), (¯ x, g(¯ x)) − (xn , yn )⟩ = 0.

n→∞

n→∞

(49)

Since CG(¯ x, 0) is metrically subregular at (0, 0) there exist κ, δ > 0 such that for every u ∈ B(0, δ) one has d(u, CG(¯ x, 0)−1 (0)) ≤ κd(0, CG(¯ x, 0)(u)), or equivalently, d(u, C ) ≤ κd(Dg(¯ x)(u), CF (¯ x, g(¯ x))(u))

Huynh and Phan: Metric subregularity of multifunctions c 0000 INFORMS Mathematics of Operations Research 00(0), pp. 000–000, ⃝

19

(since CG(¯ x, 0)(u) = Dg(¯ x)(u) − CF (¯ x, g(¯ x))(u)). Hence, for each n sufficiently large, there exist un ∈ C ∩ SX , tn ≥ 0 such that ∥xn − x ¯∥2 ∥xn − x ¯ − tn un ∥ ≤ κd(Dg(¯ x)(xn − x ¯), CF (¯ x, g(¯ x))(xn − x ¯)) + n ∥xn − x ¯ ∥2 ≤ κd(Dg(¯ x)(xn − x ¯), F (xn ) − g(¯ x)) + n (since F (xn ) − g(¯ x) ⊂ CF (¯ x, g(¯ x))(xn − x ¯)) ∥xn − x ¯∥2 = κd(g(¯ x) + Dg(¯ x)(xn − x ¯), F (xn )) + . n Then by Assumption 1 one has ∥xn − x ¯ − tn un ∥ ≤ κ[d(g(xn ), F (xn )) + η ∥xn − x ¯∥2 ] +

∥xn − x ¯ ∥2 n

(50)

which together with (46) gives

[ ]

xn − x

¯ tn ∥xn − x ¯∥ d(g(xn ), F (xn ))

− un ≤ κ + η ∥xn − x ¯∥ + → 0(n → ∞)

∥xn − x ¯∥ ∥xn − x ¯∥ ∥xn − x ¯∥ n Hence,

tn → 1. ∥xn − x ¯∥

We have the following estimations: ⟨(x∗n ,yn∗ ), (¯ x + tn un , g(¯ x + tn un ))⟩ − σgphF (x∗n , yn∗ ) = ⟨x∗n , x ¯ + tn un ⟩ + ⟨yn∗ , g(¯ x + tn un )⟩ − ⟨(x∗n , yn∗ ), (xn , yn )⟩(since x∗n ∈ D∗ F (xn , yn )(−yn∗ )) ∗ = ⟨xn , x ¯ + tn un − xn ⟩ + ⟨yn∗ , g(¯ x + tn un ) − yn ⟩ ∗ = ⟨xn , x ¯ + tn un − xn ⟩ + ⟨yn∗ , g(¯ x + tn un ) − g(xn )⟩ + ⟨yn∗ , g(xn ) − yn ⟩ ∗ ≥ ⟨xn , x ¯ + tn un − xn ⟩ + ⟨Dg(¯ x)∗ yn∗ , x ¯ + tn un − xn ⟩ − ηmax{∥tn un ∥, ∥xn − x ¯∥}∥x ¯ + tn un − xn ∥+ + (1 − ϵn )∥g(xn ) − yn ∥ (by Assumption 1 and (48)) ≥ (1 − ϵn )d(g(xn ), F (xn )) − ∥x∗n + Dg(¯ x)∗ yn∗ ∥.∥x ¯ + tn un − xn ∥ − ηn ∥x ¯ + tn un − xn ∥ (ηn := −ηmax{∥tn un ∥, ∥xn − x ¯∥} → 0) = (1 − ϵn )d(g(xn ), F (xn )) − δn ∥x ¯ + tn un − xn ∥(δn := ∥x∗n + Dg(¯ x)∗ yn∗ ∥ + ηn → 0 by (46)) 1 ≥ (1 − ϵn − κδn )d(g(xn ), F (xn )) − (ηκ + )δn ∥x ¯ − xn ∥2 (by (50)). n Therefore,

1 tn 2

[⟨(x∗n , yn∗ ),(¯ x + tn un , g(¯ x + tn un ))⟩ − σgph F (x∗n , yn∗ )] (1 − ϵn − κδn ) 1 ≥ d(g(xn ), F (xn )) − (ηκ + )δn 2 n tn

(

∥x ¯ − xn ∥ tn

)2 .

Hence,

1 [⟨(x∗n , yn∗ ), (¯ x + tn un , g(¯ x + tn un ))⟩ − σgphF (x∗n , yn∗ )] ≥ 0. tn 2 On the other hand, since lim inf n→∞

(51)

tn 2 σA(un ) (x∗n , yn∗ ) 2 tn 2 = sup ⟨(x∗n , yn∗ ), (¯ x, g(¯ x)) + tn (un , Dg(¯ x)(un )) + (w1 , w2 )⟩ 2 (w1 ,w2 )∈A(un ) ≤ σgph F (x∗n , yn∗ ) + tn 2 ξ ∥x∗n + Dg(¯ x)∗ yn∗ ∥ + ◦(tn 2 ) tn 2 (w1 , w2 ) ∈ gph F + tn 2 ξ(IX , Dg(¯ x))BX + ◦(tn 2 )BX×Y ), (since (¯ x, g(¯ x)) + tn (un , Dg(¯ x)(un )) + 2

⟨(x∗n , yn∗ ), (¯ x + tn un , g(¯ x) + tn Dg(¯ x)(un ))⟩ +

Huynh and Phan: Metric subregularity of multifunctions c 0000 INFORMS Mathematics of Operations Research 00(0), pp. 000–000, ⃝

20

then

⟨(x∗n , yn∗ ), (¯ x + tn un , g(¯ x + tn un ))⟩ − σgphF (x∗n , yn∗ ) ≤ ∗ ∗ ≤ ⟨(xn , yn ), (¯ x + tn un , g(¯ x + tn un ))⟩ − ⟨(x∗n , yn∗ ), (¯ x + tn un , g(¯ x) + tn Dg(¯ x)(un ))⟩ 2 tn σA(un ) (x∗n , yn∗ ) + tn 2 ξ ∥x∗n + Dg(¯ x)∗ yn∗ ∥ + ◦(tn 2 ) − 2 tn 2 = ⟨yn∗ , g(¯ x + tn un ) − g(¯ x) − tn Dg(¯ x)(un ))⟩ − σA(un ) (x∗n , yn∗ ) 2 + tn 2 ξ ∥x∗n + Dg(¯ x)∗ yn∗ ∥ + ◦(tn 2 ).

Therefore, 2

[⟨(x∗ , y ∗ ), (¯ x + tn un , g(¯ x + tn un ))⟩ − σgph F (x∗n , yn∗ )] ≤ tn 2⟨ n n ⟩ 2 ∗ x + tn un ) − g(¯ x) − tn Dg(¯ x)(un )] − σA(un ) (x∗n , yn∗ )+ ≤ yn , 2 [g(¯ tn ◦(tn 2 ) + 2ξ ∥x∗n + Dg(¯ x)∗ yn∗ ∥ + tn 2 2 x + tn un ) − g(¯ x) − tn Dg(¯ x)(un )]− = ⟨yn∗ , g ′′ (¯ x, un )⟩ − σA(un ) (x∗n , yn∗ ) + ⟨yn∗ , 2 [g(¯ tn 2 ◦(tn ) − g ′′ (¯ x, un )⟩ + 2ξ ∥x∗n + Dg(¯ x)∗ yn∗ ∥ + tn 2 which together with (46), (49), (41) and Assumption 2 imply lim inf n→∞

2 tn 2

[⟨(x∗n , yn∗ ), (¯ x + tn un , g(¯ x + tn un ))⟩ − σgph F (x∗n , yn∗ )] < 0

which contradicts to (51). 2. By Lemma 3 there exists ξ > 0 such that A(u) := {(0, g ′′ (¯ x, u))}, u ∈ C ∩ SX is an inner second order approximation map for gph F at (¯ x, g(¯ x)) with respect to (IX , Dg(¯ x)), ξ. Then (43) holds immediately. The last assertion of Theorem 3 is obvious from Proposition 3. The proof is complete.  Corollary 3. Let the assumptions as in the preceding theorem be satisfied. Assume further that gph F − {(¯ x, g(¯ x))} is a cone. If there exists a sequence {(xn ∗ , yn ∗ )} ⊂ X ∗ × SY ∗ satisfying lim [⟨(xn ∗ , yn ∗ ), (¯ x, g(¯ x))⟩ − σgph F (xn ∗ , yn ∗ )] = lim ∥Dg(¯ x)∗ yn ∗ + xn ∗ ∥ = 0,

n→∞

and

n→∞

x, u)⟩ > 0, lim inf sup ⟨yn ∗ , g ′′ (¯

(52)

n→∞ u∈C∩S X

then G fails to be metrically subregular at (¯ x, 0). Proof. As gph F − {(¯ x, g(¯ x))} is a cone, for all u ∈ C , all t ≥ 0, (¯ x + tu, g(¯ x) + tDg(¯ x)(u)) ∈ gph F, and therefore relation (42) holds trivially. Let A be an inner second order approximation map for gph F at (¯ x, g(¯ x)) with respect to A := (IX , Dg(¯ x)). Let {(xn ∗ , yn ∗ )} ⊂ X ∗ × SY ∗ be a sequence satisfying the relations in the corollary. For u ∈ C ∩ SX , and for any w := (w1 , w2 ) ∈ A(u), one has (¯ x, g(¯ x)) + t(u, Dg(¯ x)(u)) +

t2 w ∈ gph F + t2 ξABX + ◦(t2 ) as t → 0+ . 2

Take a sequence (tn ) of positives such that lim

1

n→∞ t2 n

[⟨(xn ∗ , yn ∗ ), (¯ x, g(¯ x))⟩ − σgph F (xn ∗ , yn ∗ )] = lim

1

n→∞ tn

∥Dg(¯ x)∗ yn ∗ + xn ∗ ∥ = 0.

Huynh and Phan: Metric subregularity of multifunctions c 0000 INFORMS Mathematics of Operations Research 00(0), pp. 000–000, ⃝

21

Then, one has ⟨(x∗n , yn∗ ), (w1 , w2 )⟩ ≤

2 [σgph F (xn ∗ , yn ∗ ) − ⟨(xn ∗ , yn ∗ ), (¯ x, g(¯ x))⟩]− t2 n 2 2 ∗ ∗ ∗ ∗ x) yn + xn ) (u) + 2ξ ∥Dg(¯ x) yn ∗ + xn ∗ ∥ + ◦(tt2n ) − tn (Dg(¯ n x, g(¯ x))⟩]+ ≤ t22 [σgph F (xn ∗ , yn ∗ ) − ⟨(xn ∗ , yn ∗ ), (¯ n 2 +2(ξ + 1/tn )∥Dg(¯ x)∗ yn ∗ + xn ∗ ∥ + ◦(tt2n ) → 0 as n → ∞. n

.

Hence, lim inf sup {⟨yn ∗ , g ′′ (¯ x, u)⟩ − σA(u) (xn ∗ , yn ∗ )} ≥ lim inf sup ⟨yn ∗ , g ′′ (¯ x, u)⟩ > 0. n→∞ u∈C∩S X

n→∞ u∈C∩S X

The conclusion follows from the second part of Theorem 3.



Now let Y be the direct sum of two Banach spaces Y1 , Y2 . Of course, we can view Y as the cartesian product of Y1 , Y2 : Y = Y1 × Y2 . Assume that g(x) := (g1 (x), g2 (x));

F (x) = (F1 (x), F2 (x)), G(x) := (G1 (x), G2 (x)) x ∈ X,

where gi : X → Yi are continuously differentiable and Fi : X ⇒ Yi are closed convex multifunctions and Gi := gi − Fi , i = 1, 2. With a proof almost similar to the one of Theorem 3, we obtain the following theorem, which generalizes Theorem 5.4 in [25]. Theorem 4. Suppose that X, Y = Y1 × Y2 are Banach spaces and Assumptions 1 and 2 are fulfilled. Assume further that the multifunction G2 is metrically subregular at (¯ x, 0) and x ¯∈ int(Dom F1 ). If the contingent derivative CG(¯ x, 0) is metrically subregular at (0, 0) and there are real ξ ≥ 0 and an inner second order approximation map A for gph F at (¯ x, g(¯ x)) with respect to (IX , Dg(¯ x)) ∗ ∗ and ξ such that for any sequences {x∗n } ⊆ X ∗ , {yn∗ } = {(yn1 , yn2 )} ⊂ (Y1∗ × Y2∗ ) ∩ SY ∗ satisfying lim [⟨(xn ∗ , yn ∗ ), (¯ x, g(¯ x))⟩ − σgph F (xn ∗ , yn∗ )] = lim ∥Dg(¯ x)∗ yn ∗ + xn ∗ ∥ = 0

n→∞

one has

n→∞

lim inf sup {⟨yn ∗ , g ′′ (¯ x, u)⟩ − σA(u) (xn ∗ , yn ∗ ) + γ ∥yn1∗ ∥} ≤ 0, n→∞ u∈C∩S X

(53)

for some γ > 0, then G is metrically subregular at (¯ x, 0). Proof. As in the proof of the previous theorem, suppose on the contrary that G is not metrically subregular at (¯ x, 0). Then by Theorem 2, there exist sequences xn → x ¯, yn = (yn1 , yn2 ) ∈ F1 (xn ) × F2 (xn ), ϵn → 0+ , and yn∗ = (yn1∗ , yn2∗ ) ∈ SY ∗ , x∗n ∈ D∗ F (xn , yn )(−yn∗ ) such that g(xn ) ∈ / F (xn ),

d(0, g(xn ) − F (xn )) → 0, ∥Dg(¯ x)∗ yn∗ + x∗n ∥ → 0, ∥xn − x ¯∥

∥gi (xn ) − yni ∥ ≤ (1 + ϵn )d(0, gi (xn ) − Fi (xn )) (i = 1, 2), |⟨yn∗ , g(xn ) − yn ⟩ − ∥g(xn ) − yn ∥| ≤ ϵn ∥g(xn ) − yn ∥. 1∗ i∗ ∗ i i∗ 2∗ Set x∗n = x1∗ n + xn with xn ∈ D Fi (xn , yn )(−yn ) (i = 1, 2). If limn→∞ ∥yn ∥ = 0, then by the assump1∗ tion x ¯ ∈ int(Dom F1 ), one has limn→∞ ∥xn ∥ = 0. Therefore, by the preceding relations yield the following d(0, g2 (xn ) − F2 (xn )) g2 (xn ) ∈ / F2 (xn ), → 0, ∥Dg2 (¯ x)∗ yn2∗ + xn2∗ ∥ → 0, ∥xn − x ¯∥

Huynh and Phan: Metric subregularity of multifunctions c 0000 INFORMS Mathematics of Operations Research 00(0), pp. 000–000, ⃝

22

∥g2 (xn ) − yn2 ∥ ≤ (1 + ϵn )d(0, g2 (xn ) − F2 (xn )), |⟨yn2∗ , g2 (xn ) − yn2 ⟩ − ∥g2 (xn ) − yn2 ∥| ≤ ϵn ∥g2 (xn ) − yn2 ∥.

By virtue of Theorem 2, the later relations contradict the metric subregularity of G2 . Hence, one has lim inf n→∞ ∥yn1∗ ∥ > 0. Finally, similarly to the proof of Theorem 3, we get a contradiction.  Remark 3. Theorem 4 above is a generalization of Theorem 5.4 in [25], in which the set valued map F is assumed to be a constant map. The main idea of the proof of Theorem 5.4 in [25] is based on an iterative scheme of Newton type. Our proof is directly based on the general sufficient condition in terms of coderivatives for metric subregularity (Theorem 2). It seems that this proof is less involved than the one in [25]. When Y is finite dimensional, one can simplify the second order condition in the preceding theorem as follows. Corollary 4. Let Y be finite dimensional and suppose that Assumptions 1,2 are fulfilled. (i) Assume that the contingent derivative CG(¯ x, 0) is metrically subregular at (0, 0). Suppose further that there are reals r > 0, ξ ≥ 0 and an inner second order approximation map A for gph F at (¯ x, g(¯ x)) with respect to (IX , Dg(¯ x)) and ξ such that for each y ∗ ∈ SY ∗ satisfying ⟨(−Dg(¯ x)∗ y ∗ , y ∗ ), (¯ x, g(¯ x))⟩ = σgph F (−Dg(¯ x)∗ y ∗ , y ∗ ),

one has

sup {⟨y ∗ , g ′′ (¯ x, u)⟩ − σA(u) (−Dg(¯ x)∗ y ∗ , y ∗ )} < 0,

(54) (55)

u∈C∩SX

and

σA(u) ((−Dg(¯ x)∗ y ∗ , y ∗ )) = σA(u)∩rBX×Y ((−Dg(¯ x)∗ y ∗ , y ∗ )), ∀u ∈ C ∩ SX .

(56)

Then G is metrically subregular at (¯ x, 0). (ii) Assume that gph F − (¯ x, g¯(¯ x)) is a cone. If there exists y ∗ ∈ SY ∗ such that ⟨(−Dg(¯ x)∗ y ∗ , y ∗ ), (¯ x, g(¯ x))⟩ = σgph F (−Dg(¯ x)∗ y ∗ , y ∗ ),

and

sup ⟨y ∗ , g ′′ (¯ x, u)⟩ > 0, u∈C∩SX

then G fails to be metrically regular at (¯ x, 0). Proof. Let {(x∗n , yn∗ )} ⊂ X ∗ × SY ∗ be a sequence satisfying relation lim [⟨(xn ∗ , yn ∗ ), (¯ x, g(¯ x))⟩ − σgph F (xn ∗ , yn∗ )] = lim ∥Dg(¯ x)∗ yn ∗ + xn ∗ ∥ = 0

n→∞

n→∞

in Theorem 4. Since Y is finite dimensional, without loss of generality, we may assume that yn∗ → x)∗ (y ∗ ), and moreover, y ∗ ∈ SY ∗ . It implies that x∗n → −Dg(¯ ⟨(−Dg(¯ x)∗ y ∗ , y ∗ ), (¯ x, g(¯ x))⟩ = lim σgph F (x∗n , yn∗ ). n→∞

Now, since σgph F is lower semicontinuous, then lim σgph F (x∗n , yn∗ ) ≥ σgph F (−Dg(¯ x)∗ y ∗ , y ∗ ).

n→∞

As (¯ x, g(¯ x)) ∈ gph F, then ⟨(−Dg(¯ x)∗ y ∗ , y ∗ ), (¯ x, g(¯ x))⟩ ≤ σgph F (−Dg(¯ x)∗ y ∗ , y ∗ );

Huynh and Phan: Metric subregularity of multifunctions c 0000 INFORMS Mathematics of Operations Research 00(0), pp. 000–000, ⃝

this implies that

23

⟨(−Dg(¯ x)∗ y ∗ , y ∗ ), (¯ x, g(¯ x))⟩ = σgph F (−Dg(¯ x)∗ y ∗ , y ∗ ).

By (56), one has sup {⟨yn∗ , g ′′ (¯ x, u)⟩ − σA(u) (x∗n , yn∗ )} ≤

u∈C∩SX

≤ sup {⟨y ∗ , g ′′ (¯ x, u)⟩ − σA(u) (−Dg(¯ x)∗ y ∗ , y ∗ )} + r(∥(x∗n , yn∗ ) − (−Dg(¯ x)∗ y ∗ , y ∗ )∥). u∈C∩SX

Therefore, from (55), one obtains lim inf sup {⟨yn∗ , g ′′ (¯ x, u)⟩ − σA(u) (x∗n , yn∗ )} < 0. n→∞ u∈C∩S X

The conclusion (i ) follows from Theorem 3. Similarly, (ii ) follows from Corollary 3.



We end the paper by giving some illustrative examples. Example 2. Let g : R3 → R2 , F : R3 ⇒ R2 and G : R3 ⇒ R2 defined by g(x1 , x2 , x3 ) = (x41 + x22 − 2x23 , −x31 + 2x22 + 3x23 ), (x1 , x2 , x3 ) ∈ R3 ; F (x1 , x2 , x3 ) = {(y1 , y2 ) ∈ R2 : x1 ≤ y1 , −x1 ≤ y2 }, (x1 , x2 , x3 ) ∈ R3 ; G(x) := g(x) − F (x),

x = (x1 , x2 , x3 ) ∈ R3 .

Obviously, gph F is a convex cone and therefore at (¯ x, 0) = ((0, 0, 0), (0, 0)) ∈ R3 × R2 , Tgph F (0R3 , 0R2 ) = gph F. Moreover, since Dg(0R3 ) = 0, then A := gph F is an inner second order approximation for gph F at (¯ x, g(¯ x)) with respect to (IX , Dg(¯ x)) (¯ x = 0R3 , g(¯ x) = 0R2 ). One has C = {u = (0, λ2 , λ3 ) : λ2 , λ3 ∈ R},

and for u = (0, λ2 , λ3 ) ∈ C ,

g ′′ (0R3 , u) = (λ21 − 2λ23 , 2λ21 + 3λ23 ).

Let y ∗ = (y1 , y2 ) ∈ SR2 be such that (54) is verified. Then, σgph F (−Dg(¯ x)∗ y ∗ , y ∗ ) = σgph F (0, y ∗ ) = 0. A direct calculation gives y1 = y2 := t < 0. Thus sup {⟨y ∗ , g ′′ (¯ x, u)⟩ − σA(u) (−Dg(¯ x)∗ y ∗ , y ∗ )}

u∈C∩SR3

= sup {t(3λ22 + λ23 )} = t < 0, u∈C∩SR3

showing the metric subregularity of G at (0R3 , 0R2 ) by Corollary 4-(i). Now, let g : R3 → R2 be defined by g(x1 , x2 , x3 ) = (x41 + x22 − 2x23 , −x31 + 2x22 + x23 ), (x1 , x2 , x3 ) ∈ R3 . For F as above, x ¯ := 0R3 , for y ∗ = (t, t) with t < 0; u := (0, λ2 , λ3 ) ∈ C , one has sup ⟨y ∗ , g ′′ (¯ x, u)⟩ = sup {t(3λ22 − λ23 )} = −t > 0.

u∈C∩SR3

u∈C∩SR3

According to Corollary 4-(ii), in this case, G is not metrically regular at (0R3 , 0R2 ).

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5. Concluding Remarks 1. The sufficient conditions established in the paper are given for the local version of metric subregularity. For the global version of metric subregularity/regularity, several conditions in terms of the strong slopes have been established in the current literature (see, e.g., [28, 51]). It is easy to see that the first part of Theorem 1 yields directly a sufficient condition for global metric subregularity of the multifunction F as follows. If for some (¯ x, y¯) ∈ gph F, there exist reals m and δ > 0 such that |∇|φ(x) ≥ m

for all x ∈ X, x ̸= x ¯

with

φ(x) < δ, d(x, x ¯)

then there is τ > 0 such that d(x, F −1 (¯ y )) ≤ τ d(y, F (¯ x))

for all x ∈ X.

However, it is not clear for us how to derive effective conditions in terms of coderivatives or contingent derivatives at points in the graph of the multifunction under consideration for global metric subregularity. Deep studies are needed to deal with this problem. 2. Recently, H¨olderian error bound properties, as well as H¨olderian metric subregularity/regularity has attracted the study of a large number of authors (see [23, 31, 40, 59] and the references given therein). Note that if we replace the function φ in relation (13) of Theorem 1 by φγ with some given γ > 0, we obtain a sufficient condition for metric subregularity with order γ of the multifunction F. We hope that by using this remark, and further examinations, some established sufficient conditions can be modified in order to obtain the ones for metric subregularitiy with positive order. Acknowledgement. The authors are grateful to the anonymous referees. many valuable comments and suggestions were helpful to improve the presentation of the paper. In particular, assumption (42), weaker than the one in the first version of the paper was suggested by one of the referees. The concluding remarks are based on the remarks of the other referee. The authors would like to thank Professor M. Th´era and N. H. Tron for the useful comments. This paper was written during the stay of the authors at VIASM in Summer 2012. We wish to express our thanks to the Institute for support, especially to Professor Le Tuan Hoa, the former director of VIASM, and the secretaries of the Institute for hospitality and thoughtfulness. References [1] Arag´on Artacho F. J., Mordukhovich B.S 2010, Metric regularity and Lipschitzian stability of parametric variational systems, Nonlinear Analysis, 72, 1149-1760. [2] Aubin J.-P 1984, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res., 9, 87-111. [3] Aussel D., Corvellec J.-N., Lassonde M. 1995, Mean value property and subdifferential criteria for lower semicontinuous functions., Trans. Amer. Math. Soc., 347, no. 10, 41474161. [4] Az´e D. 2003, A survey on error bounds for lower semicontinuous functions, Proceedings of 2003 MODESMAI Conference, ESAIM Proc., 1, 1–17, EDP Sciences. [5] Az´e D. 2006, A unified theory for metric regularity of multifunctions, Journal of Convex Anal., 13, no. 2, 225-252. [6] Bonnans J. F., Shapiro A. 2000, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York. [7] Borwein J. M, Dontchev A. L. 2003, On the Bartle-Graves theorem. Proc. Amer. Math. Soc. 131, no. 8, 2553–2560. [8] Borwein J. M., Zhu Q. J. 1996, Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity. SIAM J. Control Optim., 34, no. 5, 1568–1591.

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