We establish formulas for computingestimating the Fr´echet and Mordukhovich coderivatives of implicit multifunctions defined by generalized equations in Asplund spaces. These formulas are applied to obtain conditions for solution stability of parametric variational systems over perturbed smoothboundary constraint sets.
Trang 1Coderivatives of Implicit Multifunctions and
June 22, 2015
Abstract We establish formulas for computing/estimating the Fr´echet and Mordukhovich coderivatives of implicit multifunctions defined by generalized equations in Asplund spaces These formulas are applied to obtain conditions for solution stability of parametric variational systems over perturbed smooth-boundary constraint sets
Key words Coderivative, generalized equation, implicit multifunction, local Lipschitz-like property, normal cone operator, variational system
AMS subject classification 49J53, 49J52, 49J40
1 Introduction
It is well-known that the classical implicit function theorem (see, e.g., [4, Theorem 2.1] or [22, Theorem 9.28]) is one of important theorems in mathematical analysis It gives a formula for computing the Fr´echet derivative of an implicit function at a given point in its effective domain Using the obtained formula one can find the adjoint of the Fr´echet derivative of the implicit function, which is called the coderivative (a kind of generalized derivatives) of the implicit function It can be seen in [2] that implicit function theorems are used extensively
in numerical analysis and solution stability theory
In accordance with developing of variational analysis, nonsmooth analysis, and set-valued analysis, implicit multifunctions appear naturally as extensions of implicit functions Gen-eralized differentiation properties of implicit multifunctions allow one to obtain interesting results on solution sensitivity/stability of (parametric) variational systems (see the references cited below) In variational analysis, the Fr´echet coderivative and the normal coderivative (also called the Mordukhovich coderivative) of multifunctions are well-known notions of generalized differentiation; see [10, 11, 21]
Implicit multifunctions defined by an inclusion involving a single-valued function and a multifunction (generalized equation in the terminology of Robinson [20]) have been consid-ered by many researchers In [8], Levy and Mordukhovich obtained an upper estimate for the values of the normal coderivative of the implicit multifunctions in finite dimensional spaces Later, Lee and Yen [6] provided us with formulas for computing/estimating coderivatives of the implicit multifunctions Furthermore, Mordukhovich [9] established an upper estimate for the values of the normal coderivative of the implicit multifunctions, in a Banach space
∗ This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.56.
† College of Information and Communication Technology, Can Tho University, Campus II, 3/2 Street, Can Tho, Vietnam; Email: ntqui@cit.ctu.edu.vn; ntqui.vn@gmail.com.
Trang 2setting, under the assumption that the multifunction in the inclusion has the SNC property (see the definition in the next section) Related to the topic of coderivatives of implicit multifunctions in Banach spaces, a lower estimate for the values of the Fr´echet coderivative
of implicit multifunctions under a select of assumptions can be found in [5]
The results of Lee and Yen in [6] on coderivatives of implicit multifunctions are applied to establish verifiable necessary and sufficient conditions for the local Lipschitz-like property and the local metric regularity in Robinson’s sense of the solution maps of parametric variational inequalities over unperturbed/perturbed polyhedral convex sets or smooth-boundary sets
in finite dimensional spaces; see e.g., [6, 7, 14, 15, 16, 18, 17, 19] Concerning solution stability of variational inequalities/systems using coderivative criteria, we refer the reader
to [1, 3, 12, 23, 24, 25] for more details Note that, in the paper [25], the smooth-boundary constraint sets of variational systems are fixed
In this paper, we focus on implicit multifunctions defined by inclusions involving a single-valued function and a multifunction Analyzing and developing furthermore the aforemention upper estimate for the values of the Mordukhovich coderivative of the implicit multifunc-tions in [9], we obtain formulas for computing/estimating the Fr´echet coderivative and the Mordukhovich coderivative of the implicit multifunctions in a class of infinite dimensional spaces; called Asplund spaces An important assumption for our investigation is that the multifunctions in the inclusions defined the implicit multifunctions have the SNC property
On the basis of the explicit formulas of coderivatives of the normal cone operator to parametric smooth-boundary sets in Asplund spaces given in [13], the coderivative formulas
of implicit multifunctions are used to derive solution stability criteria of parametric varia-tional systems given by inclusions involving a C1-function and the normal cone operator to
a smooth-boundary constraint set in finite dimensional spaces When the constraint sets are convex, the parametric variational systems are termed parametric variational inequali-ties over perturbed smooth-boundary sets We will show that the normal cone operator to parametric smooth-boundary sets may not be SNC in infinite dimensional Asplund spaces (such as the Hilbert space `2); see Lemma 4.1 Hence, it is important to stress that we cannot establish solution stability criteria for the above parametric variational systems via the coderivative formulas of the normal cone operator in infinite dimensional spaces because
of the surprising property of the operator
The rest of this paper is organized as follows Section 2 recalls some basic concepts and facts from variational analysis Section 3 provides exact formulas for computing the Fr´echet and Mordukhovich coderivatives of the implicit multifunctions in Asplund spaces Section 4 establishes criteria for stability of parametric variational systems Several examples are also given in this section
2 Preliminaries
Let F : X ⇒ Y be a multifunction between Banach spaces The graph and kernel of F are gphF := {(x, y) ∈ X × Y | y ∈ F (x)} and ker F := {x ∈ X| 0 ∈ F (x)} respectively A set
Ω ⊂ X is locally closed around x ∈ Ω if there exists a closed ball in X centered at x with radius ε > 0, denoted by ¯B(x, ε), such that Ω ∩ ¯B(x, ε) is closed One says that F is locally closed around (¯x, ¯y) ∈ gphF if gphF is locally closed around (¯x, ¯y) in the product space
X × Y Unless otherwise stated, every norm in question of a product normed space is the sum norm
The sequential Painlev´e-Kuratowski upper/outer limit of F as x → ¯x is defined by
Limsup
x→¯ x
F (x) =y ∈ Y
∃xk→ ¯x and yk→ y with yk ∈ F (xk), ∀k ∈ IN , (2.1)
Trang 3where IN := {1, 2, } and the limits are understood in the sequential sense.
Let Ω ⊂ X with ¯x ∈ Ω The set T (¯x; Ω) ⊂ X defined by
T (¯x; Ω) := Limsup
t↓0
Ω − ¯x
where “Limsup” is taken with respect to the norm topology of X, is the Bouligand-Severi tangent/contingent cone to Ω at ¯x From (2.2) and (2.1) we have
T (¯x; Ω) =z ∈ X| ∃tk ↓ 0, ∃zk → z with ¯x + tkzk ∈ Ω, ∀k ∈ IN (2.3) The set of ε-normals to Ω at ¯x ∈ Ω is given by
b
Nε(¯x; Ω) :=
x∗ ∈ X∗
limsup
x →¯Ωx
hx∗, x − ¯xi
kx − ¯xk ≤ ε
where x → ¯Ω x means x → ¯x with x ∈ Ω When ε = 0, the set in (2.4) is called the Fr´echet normal cone to Ω at ¯x and denoted by bN (¯x; Ω) If ¯x 6∈ Ω, one puts bNε(¯x; Ω) = ∅ for all ε ≥ 0
by convention The limiting/Mordukhovich normal cone to Ω at ¯x ∈ Ω is the set
N (¯x; Ω) := Limsup
x→¯ x ε↓0
b
Nε(x; Ω), (2.5)
where “Limsup” is taken with respect to the norm topology of X and the weak* topology
of X∗ By convention, one puts N (¯x; Ω) = ∅ when ¯x 6∈ Ω
A Banach space X is Asplund if every convex continuous function ϕ : U → IR defined on
an open convex subset U of X is Fr´echet differentiable on a dense subset of U The class of Asplund spaces is large For instance, it contains the class of reflexive Banach spaces The calculus of normal cones in Asplund spaces is simpler than that in general Banach spaces Namely, when X is an Asplund space and Ω ⊂ X is locally closed around ¯x ∈ Ω, the limiting normal cone to Ω at ¯x defined by (2.5) is computed by the formula
N (¯x, Ω) = Limsup
x→¯ x
b
Given (¯x, ¯y) ∈ gphF , we call the multifunction bD∗εF (¯x, ¯y) : Y∗ ⇒ X∗ defined by
b
D∗εF (¯x, ¯y)(y∗) =x∗ ∈ X∗
(x∗, −y∗) ∈ bNε((¯x, ¯y); gphF ) , ∀y∗ ∈ Y∗, (2.7)
the ε-coderivative of F at (¯x, ¯y) When ε = 0, bD∗εF (¯x, ¯y) is called the Fr´echet coderivative
of F at (¯x, ¯y) and it is denoted by bD∗F (¯x, ¯y) The multifunction D∗F (¯x, ¯y) : Y∗ ⇒ X∗,
D∗F (¯x, ¯y)(y∗) = x∗ ∈ X∗
(x∗, −y∗) ∈ N ((¯x, ¯y); gphF ) , ∀y∗ ∈ Y∗, (2.8)
is said to be the Mordukhovich coderivative of F at (¯x, ¯y) From (2.6), (2.7), and (2.8) we see that if X and Y are Asplund spaces and gphF is locally closed around (¯x, ¯y), then
D∗F (¯x, ¯y)(y∗) = Limsup
(x,y)→(¯ x,¯ y)
z∗w
∗
→y ∗
b
D∗F (x, y)(z∗), ∀y∗ ∈ Y∗ (2.9)
The multifunction F is graphically regular at (¯x, ¯y) if D∗F (¯x, ¯y)(y∗) = bD∗F (¯x, ¯y)(y∗) for all
y∗ ∈ Y∗ The last condition is equivalent to that N ((¯x, ¯y); gphF ) = bN ((¯x, ¯y); gphF ) If F
Trang 4is a single-valued mapping and ¯y = F (¯x), we write bD∗F (¯x) and D∗F (¯x) for bD∗F (¯x, ¯y) and
D∗F (¯x, ¯y) respectively
One says that F is locally Lipschitz-like (or, in the terminology of [2], F has the Aubin property) around (¯x, ¯y) ∈ gphF if there exist ` > 0 and neighborhoods U of ¯x and V of ¯y such that
F (x) ∩ V ⊂ F (u) + `kx − uk ¯BY, ∀x, u ∈ U, where ¯BY is the closed unit ball of Y
A multifunction F : X ⇒ Y is sequentially normally compact (SNC) at (¯x, ¯y) ∈ gphF if for any sequence (εk, xk, yk, x∗k, yk∗) ∈ [0, ∞) × (gphF ) × X∗× Y∗ satisfying
εk ↓ 0, (xk, yk) → (¯x, ¯y), x∗k ∈ bD∗ε
kF (xk, yk)(y∗k) (2.10) one has
(x∗k, y∗k)→ (0, 0) =⇒ k(xw∗ ∗k, y∗k)k → 0 as k → ∞
The multifunction F is partially sequentially normally compact (PSNC) at (¯x, ¯y) ∈ gphF if for any sequence (εk, xk, yk, x∗k, yk∗) ∈ [0, ∞) × (gphF ) × X∗× Y∗ satisfying (2.10) one has
h
x∗k → 0 and kyw∗ k∗k → 0i =⇒ kx∗kk → 0 as k → ∞
When both X and Y are Asplund and gphF is locally around (¯x, ¯y), we can put εk = 0 in the above properties
3 Coderivatives of implicit multifunctions
In this section we establish formulas for computing/estimating the Fr´echet and Mordukhovich coderivatives of implicit multifunctions defined by generalized equations in Asplund spaces Let us consider the implicit multifunction S : X ⇒ Y defined by a generalized equation
as follows:
S(x) := {y ∈ Y | 0 ∈ f (x, y) + Q(x, y)}, (3.1) where f : X × Y → Z is a single-valued mapping and Q : X × Y ⇒ Z is a multifunction between Banach spaces The theorem below gives an upper estimate for the values of the Mordukhovich coderivative of S(·) defined by (3.1)
Theorem 3.1 (See [9, Theorem 4.1]) Consider S(·) defined by (3.1) and (¯x, ¯y) ∈ gphS, where X, Y , Z are Asplund Suppose that f : X × Y → Z is continuous around (¯x, ¯y), gphQ
is locally closed around (¯x, ¯y, ¯z) with ¯z := −f (¯x, ¯y) If Q is SNC at (¯x, ¯y, ¯z) and
h
(x∗, y∗) ∈ D∗f (¯x, ¯y)(z∗) ∩ − D∗
Q(¯x, ¯y, ¯z)(z∗) =⇒ (x∗, y∗, z∗) = (0, 0, 0), (3.2)
then one has the upper estimate
D∗S(¯x, ¯y)(y∗) ⊂nx∗ ∈ X∗
∃z∗ ∈ Z∗ with (x∗, −y∗) ∈ D∗f (¯x, ¯y)(z∗) + D∗Q(¯x, ¯y, ¯z)(z∗)o (3.3)
From now on, we suppose that X, Y , Z are Asplund spaces
We first consider the implicit multifunction (3.1) in the case that f ≡ 0, i.e., the implicit multifunction S : X ⇒ Y defined by
S(x) = {y ∈ Y | 0 ∈ Q(x, y)} (3.4)
Trang 5For every ω = (x, y) ∈ gphS, we define the sets
b Λ(Q, ω)(y∗) := [
z ∗ ∈Z ∗
n
x∗ ∈ X∗ (x∗, −y∗) ∈ bD∗Q(x, y, 0)(z∗)
o , (3.5)
and
Λ(Q, ω)(y∗) := [
z ∗ ∈Z ∗
n
x∗ ∈ X∗
(x∗, −y∗) ∈ D∗Q(x, y, 0)(z∗)o (3.6)
Coderivative formulas of S(·) defined by (3.4) are established in the following theorem
Theorem 3.2 Consider the implicit multifunction (3.4) and let ¯ω := (¯x, ¯y) ∈ gphS Then the following estimates hold
b Λ(Q, ¯ω)(y∗) ⊂ bD∗S(¯ω)(y∗) ⊂ D∗S(¯ω)(y∗), ∀y∗ ∈ Y∗ (3.7)
If Q is locally closed around (¯x, ¯y, 0), SNC at this point, and
ker D∗Q(¯x, ¯y, 0) = {0}, (3.8)
then the estimates below hold
b
Λ(Q, ¯ω)(y∗) ⊂ bD∗S(¯ω)(y∗) ⊂ D∗S(¯ω)(y∗) ⊂ Λ(Q, ¯ω)(y∗), ∀y∗ ∈ Y∗ (3.9)
If in addition Q is graphically regular at (¯x, ¯y, 0), then
b
Λ(Q, ¯ω)(y∗) = bD∗S(¯ω)(y∗) = D∗S(¯ω)(y∗) = Λ(Q, ¯ω)(y∗), ∀y∗ ∈ Y∗ (3.10) Proof The last inclusion in (3.7) is obvious due to the definitions of coderivatives We now verify the first inclusion in (3.7) For any x∗ ∈ bΛ(Q, ¯ω)(y∗), one can find z∗ ∈ Z∗ satisfying (x∗, −y∗) ∈ bD∗Q(¯x, ¯y, 0)(z∗) We show that x∗ ∈ bD∗S(¯x, ¯y)(y∗) The last inclusion means that (x∗, −y∗) ∈ bN ((¯x, ¯y); gphS), i.e.,
limsup
(x,y)gphS−→ (¯ x,¯ y)
hx∗, x − ¯xi − hy∗, y − ¯yi
kx − ¯xk + ky − ¯yk ≤ 0 (3.11)
Since (x∗, −y∗) ∈ bD∗Q(¯x, ¯y, 0)(z∗), we have (x∗, −y∗, −z∗) ∈ bN ((¯x, ¯y, 0); gphQ) The last inclusion means
limsup
(x,y,z)gphQ−→ (¯ x,¯ y,0)
hx∗, x − ¯xi − hy∗, y − ¯yi − hz∗, zi
kx − ¯xk + ky − ¯yk + kzk ≤ 0 (3.12)
For any (x, y)gphS−→ (¯x, ¯y), we have (x, y, 0)gphQ−→ (¯x, ¯y, 0) Hence, from (3.12) we obtain (3.11) which yields x∗ ∈ bD∗S(¯x, ¯y)(y∗) Therefore, (3.7) has been proved
We now suppose that Q is locally closed around (¯x, ¯y, 0), SNC at this point, and the condi-tion (3.8) holds Letting f ≡ 0, we have D∗f (¯x, ¯y)(z∗) = {(0, 0)} Hence, the condition (3.8) yields (3.2) By Theorem 3.1, the estimate D∗S(¯x, ¯y)(y∗) ⊂ Λ(Q, ¯ω)(y∗) holds, and thus we obtain (3.9) If Q is graphically regular at (¯x, ¯y, 0), then D∗Q(¯x, ¯y, 0)(z∗) = bD∗Q(¯x, ¯y, 0)(z∗) for all z∗ ∈ Z∗ From (3.9), (3.5), and (3.6) we get (3.10) 2
We now consider the implicit multifunction S(·) defined by (3.1) For ω = (x, y) ∈ gphS and (x, y, z) ∈ gphQ with z := −f (x, y), when f : X × Y → Z is continuously differentiable around ω, we define the sets
b
Λ(Q, f, ω)(y∗) := [
z ∗ ∈Z ∗
n
x∗ ∈ X∗ (x∗, −y∗) ∈ ∇f (x, y)∗z∗+ bD∗Q(x, y, z)(z∗)
o , (3.13)
Trang 6Λ(Q, f, ω)(y∗) := [
z ∗ ∈Z ∗
n
x∗ ∈ X∗ (x∗, −y∗) ∈ ∇f (x, y)∗z∗+ D∗Q(x, y, z)(z∗)
o (3.14)
We derive formulas for estimating/computing the Fr´echet and Mordukhovich coderivatives
of S(·) defined by (3.1) in the forthcoming theorem
Theorem 3.3 Consider the implicit multifunction (3.1) and let ¯ω := (¯x, ¯y) ∈ gphS and (¯x, ¯y, ¯z) ∈ gphQ with ¯z := −f (¯x, ¯y) Suppose that f : X × Y → Z is continuously differen-tiable around ¯ω Then the following estimates hold
b Λ(Q, f, ¯ω)(y∗) ⊂ bD∗S(¯ω)(y∗) ⊂ D∗S(¯ω)(y∗), ∀y∗ ∈ Y∗ (3.15)
If Q is locally closed around (¯x, ¯y, ¯z) and SNC at this point, and the constraint qualification
0 ∈ ∇f (¯x, ¯y)∗z∗+ D∗Q(¯x, ¯y, ¯z)(z∗) =⇒ z∗ = 0 (3.16)
is satisfied, then the estimates below hold
b
Λ(Q, f, ¯ω)(y∗) ⊂ bD∗S(¯ω)(y∗) ⊂ D∗S(¯ω)(y∗) ⊂ Λ(Q, f, ¯ω)(y∗), ∀y∗ ∈ Y∗ (3.17)
If, in addition, Q is graphically regular at (¯x, ¯y, ¯z), then
b
Λ(Q, f, ¯ω)(y∗) = bD∗S(¯ω)(y∗) = D∗S(¯ω)(y∗) = Λ(Q, f, ¯ω)(y∗), ∀y∗ ∈ Y∗ (3.18) Proof Define eQ(x, y) = f (x, y) + Q(x, y) By [10, Theorem 1.62], we have the equalities
b
D∗Q(¯e x, ¯y, 0)(z∗) = ∇f (¯x, ¯y)∗z∗+ bD∗Q(¯x, ¯y, −f (¯x, ¯y))(z∗), ∀z∗ ∈ Z∗,
and
D∗Q(¯e x, ¯y, 0)(z∗) = ∇f (¯x, ¯y)∗z∗+ D∗Q(¯x, ¯y, −f (¯x, ¯y))(z∗), ∀z∗ ∈ Z∗
Observe that eQ is locally closed around (¯x, ¯y, 0) Indeed, since Q is locally closed around (¯x, ¯y, ¯z), there exists ε > 0 such that ¯B (¯x, ¯y, ¯z), ε ∩ gphQ is closed From the continuity
of f one can find δ ∈ (0, ε/2] such that kf (x, y) − f (¯x, ¯y)k ≤ ε/2 for all (x, y) ∈ ¯B((¯x, ¯y), δ)
We show that G := ¯B (¯x, ¯y, 0), δ ∩ gphQ is closed Suppose that (xe k, yk, zk) ∈ G and (xk, yk, zk) → (x,b y,b z) Then we have (b x,b y,b z) ∈ ¯b B (¯x, ¯y, 0), δ In addition, using the sum norm in the product spaces we have
k(xk, yk) − (¯x, ¯y)k ≤ k(xk, yk, zk) − (¯x, ¯y, 0)k ≤ δ
Hence, kf (xk, yk) − f (¯x, ¯y)k ≤ ε/2 Since ¯z = −f (¯x, ¯y), we have
k(xk, yk, zk− f (xk, yk)) − (¯x, ¯y, ¯z)k
≤ k(xk, yk, zk) − (¯x, ¯y, 0)k + kf (xk, yk) − f (¯x, ¯y)k
≤ δ + ε/2 ≤ ε
Thus, we have (xk, yk, zk− f (xk, yk)) ∈ ¯B (¯x, ¯y, ¯z), ε ∩ gphQ and
(xk, yk, zk− f (xk, yk)) → (bx,by,bz − f (bx,by)) as k → ∞
Since ¯B (¯x, ¯y, ¯z), ε ∩gphQ is closed, (bx,by,bz −f (bx,by)) ∈ ¯B (¯x, ¯y, ¯z), ε ∩gphQ The inclusion (x,b by,bz − f (bx,y)) ∈ gphQ implies that (b x,b y,b z) ∈ gph eb Q We have shown that eQ is locally closed around (¯x, ¯y, 0)
Trang 7We see that eQ is SNC at (¯x, ¯y, 0) Indeed, taking any sequences (xk, yk, zk, x∗k, yk∗, zk∗) from gph eQ × X∗× Y∗× Z∗ with
(xk, yk, zk) → (¯x, ¯y, 0), (x∗k, y∗k) ∈ bD∗Q(xe k, yk, zk)(zk∗), (x∗k, yk∗, zk∗)→ (0, 0, 0),w∗ (3.19)
it holds that (xk, yk, zk− f (xk, yk)) ∈ gphQ, (xk, yk, zk− f (xk, yk)) → (¯x, ¯y, ¯z), and
(x∗k, y∗k) − ∇f (xk, yk)∗zk∗ ∈ bD∗Q(xk, yk, zk− f (xk, yk))(zk∗)
The last inclusion is due to (x∗k, yk∗) ∈ bD∗Q(xe k, yk, zk)(zk∗) and [10, Theorem 1.62] From (3.19) we have ((x∗k, y∗k) − ∇f (xk, yk)∗z∗k, zk∗) → (0, 0, 0) Since Q is SNC at (¯w∗ x, ¯y, ¯z), using the sum norm in the product space we have
k((x∗
k, y∗k) − ∇f (xk, yk)∗zk∗, zk∗)k = k(x∗k, y∗k) − ∇f (xk, yk)∗zk∗k + kz∗
kk → 0 as k → ∞ Hence, it holds as k → ∞ that
k(x∗k, yk∗) − ∇f (xk, yk)∗zk∗k → 0, kzk∗k → 0, k∇f (xk, yk)∗zk∗k → 0
This implies that k(x∗k, y∗k)k → 0 as k → ∞ Consequently, k(x∗k, y∗k, zk∗)k → 0 as k → ∞, which verifies the SNC property of eQ at (¯x, ¯y, 0)
The constraint qualification (3.16) is equivalent to ker D∗Q(¯e x, ¯y, 0) = {0}
Applying Theorem 3.2 to the implicit multifunction S(x) = {y ∈ Y | 0 ∈ eQ(x, y)} we get all the desired assertions of the theorem 2
Furthermore, we consider the parametric implicit multifunction S1 : X × Z ⇒ Y defined
by parametric generalized equation:
S1(x, q) = {y ∈ Y | q ∈ f (x, y) + Q(x, y)}, (3.20)
where f : X × Y → Z and Q : X × Y ⇒ Z For ω = (x, q, y) ∈ gphS1 and (x, y, z) ∈ gphQ with z := q − f (x, y), when f : X × Y → Z is continuously differentiable around ω, we put
b
Λ1(Q, f, ω)(y∗) := [
z ∗ ∈Z ∗
n (x∗, q∗) ∈ X∗× Z∗
(x∗, −y∗, q∗) ∈ ∇f (x, y)∗z∗× {−z∗} + bD∗Q(x, y, z)(z∗) × {0}
o ,
(3.21) and
Λ1(Q, f, ω)(y∗) := [
z ∗ ∈Z ∗
n (x∗, q∗) ∈ X∗× Z∗
(x∗, −y∗, q∗) ∈ ∇f (x, y)∗z∗× {−z∗} +D∗Q(x, y, z)(z∗) × {0}o
(3.22)
We will see that in coderivative formulas of S1(x, q) obtained via the sets (3.21) and (3.22), constraint qualifications as (3.16) can be removed The next theorem provides formulas for estimating/computing the Fr´echet and Mordukhovich coderivatives of S1(x, q) without using any constraint qualification as (3.16)
Theorem 3.4 Let ¯ω := (¯x, ¯q, ¯y) ∈ gphS1 with S1(·) given by (3.20) and let (¯x, ¯y, ¯z) ∈ gphQ with ¯z := ¯q − f (¯x, ¯y) Suppose that f : X × Y → Z is continuously differentiable around ¯ω Then for every y∗ ∈ Y∗ one has
b
Λ1(Q, f, ¯ω)(y∗) ⊂ bD∗S1(¯ω)(y∗) ⊂ D∗S1(¯ω)(y∗) (3.23)
Trang 8If Q is locally closed around (¯x, ¯y, ¯z), SNC at this point, then the estimates below hold
b
Λ1(Q, f, ¯ω)(y∗) ⊂ bD∗S1(¯ω)(y∗) ⊂ D∗S1(¯ω)(y∗) ⊂ Λ1(Q, f, ¯ω)(y∗) (3.24)
If, in addition, Q is graphically regular at (¯x, ¯y, ¯z), then
b
Λ1(Q, f, ¯ω)(y∗) = bD∗S1(¯ω)(y∗) = D∗S1(¯ω)(y∗) = Λ1(Q, f, ¯ω)(y∗) (3.25)
Proof Define eQ(x, y, q) = f (x, y) − q + Q(x, y) By [10, Theorem 1.62], for every z∗ ∈ Z∗ one has the following equalities
b
D∗Q(¯e x, ¯y, ¯q, 0)(z∗) = ∇f (¯x, ¯y)∗z∗× {0} − {0} × {0} × {z∗} + bD∗Q(¯x, ¯y, ¯z)(z∗) × {0}
= ∇f (¯x, ¯y)∗z∗× {−z∗} + bD∗Q(¯x, ¯y, ¯z)(z∗) × {0}, and
D∗Q(¯e x, ¯y, ¯q, 0)(z∗) = ∇f (¯x, ¯y)∗z∗× {0} − {0} × {0} × {z∗} + D∗Q(¯x, ¯y, ¯z)(z∗) × {0}
= ∇f (¯x, ¯y)∗z∗× {−z∗} + D∗Q(¯x, ¯y, ¯z)(z∗) × {0}
(3.26) Since Q is locally closed around (¯x, ¯y, ¯z), the set ¯B (¯x, ¯y, ¯z), ε ∩ gphQ is closed for some
ε > 0 Note that there exists δ ∈ (0, ε/2] such that kf (x, y) − f (¯x, ¯y)k ≤ ε/2 for every (x, y) ∈ ¯B((¯x, ¯y), δ) The set G := ¯B (¯x, ¯y, ¯q, 0), δ ∩ gphQ is closed Indeed, suppose thate (xk, yk, qk, zk) ∈ G and (xk, yk, qk, zk) → (bx,y,b p,b z) Then, (b x,b by,q,bz) ∈ ¯b B (¯x, ¯y, ¯q, 0), δ as
¯
B (¯x, ¯y, ¯q, 0), δ is closed Since k(xk, yk) − (¯x, ¯y)k ≤ k(xk, yk, qk, zk) − (¯x, ¯y, ¯q, 0)k ≤ δ, one has kf (xk, yk) − f (¯x, ¯y)k ≤ ε/2 We have ¯z = ¯q − f (¯x, ¯y), thus
k(xk, yk, zk+ qk− f (xk, yk)) − (¯x, ¯y, ¯z)k
≤ k(xk, yk, zk) − (¯x, ¯y, 0)k + kqk− ¯qk + kf (xk, yk) − f (¯x, ¯y)k
= k(xk, yk, qk, zk) − (¯x, ¯y, ¯q, 0)k + kf (xk, yk) − f (¯x, ¯y)k
≤ δ + ε/2 ≤ ε
This means that (xk, yk, zk+ qk− f (xk, yk)) ∈ ¯B (¯x, ¯y, ¯z), ε ∩ gphQ Since ¯B (¯x, ¯y, ¯z), ε ∩ gphQ is closed and (xk, yk, zk+ qk− f (xk, yk)) → (bx,by,bz +bq − f (bx,by)) as k → ∞, one has (x,b by,bz +q − f (b x,b y)) ∈ ¯b B (¯x, ¯y, ¯z), ε ∩ gphQ The inclusion (x,b y,b z +b q − f (b x,b y)) ∈ gphQb yields (x,b y,b q,b bz) ∈ gph eQ which verifies the local closedness of eQ around (¯x, ¯y, ¯q, 0)
We see that eQ is SNC at (¯x, ¯y, ¯q, 0) Indeed, taking any (xk, yk, qk, zk, x∗k, yk∗, q∗k, zk∗) in gph eQ × X∗× Y∗× Z∗× Z∗ with (xk, yk, qk, zk) → (¯x, ¯y, ¯q, 0), (x∗k, y∗k, qk∗, zk∗) w
∗
→ (0, 0, 0, 0), and (x∗k, yk∗, q∗k) ∈ bD∗Q(xe k, yk, qk, zk)(z∗k), (3.27)
one has (xk, yk, zk+ qk− f (xk, yk)) ∈ gphQ, (xk, yk, zk+ qk− f (xk, yk)) → (¯x, ¯y, ¯z), and (x∗k, yk∗, qk∗) ∈ (∇f (xk, yk)∗zk∗, −zk∗) + bD∗Q(xk, yk, zk+ qk− f (xk, yk))(zk∗) × {0} (3.28) The last inclusion is due to (3.27) and [10, Theorem 1.62] The inclusion (3.28) is equivalent to
(
(x∗k, yk∗) − ∇f (xk, yk)∗z∗k ∈ bD∗Q(xk, yk, zk+ qk− f (xk, yk))(z∗k)
qk∗ = −zk∗
By (x∗k, yk∗, q∗k, z∗k) w
∗
→ (0, 0, 0, 0), we get ((x∗
k, yk∗) − ∇f (xk, yk)∗z∗k, z∗k) w
∗
→ (0, 0, 0) Since Q is SNC at (¯x, ¯y, ¯z), using the sum norm in the product space we have
k((x∗k, y∗k) − ∇f (xk, yk)∗zk∗, zk∗)k = k(x∗k, y∗k) − ∇f (xk, yk)∗zk∗k + kzk∗k → 0 as k → ∞
Trang 9Therefore, letting k → ∞, we obtain
k(x∗k, yk∗) − ∇f (xk, yk)∗z∗kk → 0, kqk∗k = kzk∗k → 0, k∇f (xk, yk)∗zk∗k → 0
Consequently, k(x∗k, y∗k)k → 0 as k → ∞ It follows that k(x∗k, yk∗, q∗k, zk∗)k → 0 as k → ∞
We have shown that eQ is SNC at (¯x, ¯y, ¯q, 0)
We verify that ker D∗Q(¯e x, ¯y, ¯q, 0) = {0} For any z∗ ∈ Z∗ with 0 ∈ D∗Q(¯e x, ¯y, ¯q, 0)(z∗),
by (3.26) we infer that z∗ = 0 Hence, ker D∗Q(¯e x, ¯y, ¯q, 0) = {0} Applying Theorem 3.2
to the implicit multifunction S1(x, q) = {y ∈ Y | 0 ∈ eQ(x, y, q)} we obtain all the desired
4 Solution stability of variational systems
Lipschitzian stability of (parametric) variational systems are investigated in this section The coderivative formulas of implicit multifunctions are applied to construct criteria for the local Lipschitz-like property of the solution maps of the variational systems
Let us consider a C1-smooth function φ : X × Z → X∗ and a normal cone operator
F (x, p) = N (x; C(p)) with N (x; C(p)) being the normal cone in the sense of Mordukhovich
to C(p) at x Herein, C(p) is the constraint set given by
C(p) := {x ∈ X| ψ(x, p) ≤ 0}, (4.1) where ψ : X × P → IR is a C2-smooth function with P being an Asplund space For every
p ∈ P , the boundary of C(p) is the set ∂C(p) := {x ∈ X| ψ(x, p) = 0} Following [13], the values of the normal cone operator F (·) are computed by the formula:
F (x, p) =
{0}, if ψ(x, p) < 0
µ∇xψ(x, p) µ ≥ 0 , if ψ(x, p) = 0 and ∇xψ(x, p) 6= 0
∅, if ψ(x, p) > 0
(4.2)
for every (x, p) ∈ X × P
For each (¯x, ¯p, ¯v∗) ∈ X × P × X∗ with ∇xψ(¯x, ¯p) 6= 0, we define the sets
Ω1(¯x, ¯p, ¯v∗)(v∗∗) := n(x∗, p∗) ∈ X∗× P∗
x∗ = γ∇xψ(¯x, ¯p) + µ∇2
xxψ(¯x, ¯p)∗v∗∗,
p∗ = γ∇pψ(¯x, ¯p) + µ∇2xpψ(¯x, ¯p)∗v∗∗, γ ∈ IRo, (4.3)
where µ := k¯v∗k · k∇xψ(¯x, ¯p)k−1,
Ω2(¯x, ¯p, ¯v∗)(v∗∗) := IR+∇ψ(¯x, ¯p) = γ ∇xψ(¯x, ¯p), ∇pψ(¯x, ¯p) γ ∈ IR+ , (4.4) and
Ω3(¯x, ¯p, ¯v∗)(v∗∗) := IR∇ψ(¯x, ¯p) = γ ∇xψ(¯x, ¯p), ∇pψ(¯x, ¯p) γ ∈ IR , (4.5) for every v∗∗∈ X∗∗ Exact formulas for computing the Fr´echet and Mordukhovich coderiva-tives of F (·) are as follows
Theorem 4.1 (See [13, Theorems 3.2 and 3.3]) For any (¯x, ¯p, ¯v∗) ∈ gphF , the following assertions are valid:
(i) If ψ(¯x, ¯p) < 0, then ¯v∗ = 0 and
D∗F (¯x, ¯p, ¯v∗)(v∗∗) = bD∗F (¯x, ¯p, ¯v∗)(v∗∗) = {(0X∗, 0P∗)}, ∀v∗∗ ∈ X∗∗ (4.6)
Trang 10(ii) If ψ(¯x, ¯p) = 0, ∇xψ(¯x, ¯p) 6= 0, ∇pψ(¯x, ¯p) 6= 0, and ¯v∗ 6= 0, then ¯v∗ = µ∇xψ(¯x, ¯p) for some µ > 0 In this case, for every v∗∗ ∈ X∗∗, one has
D∗F (¯x, ¯p, ¯v∗)(v∗∗) = bD∗F (¯x, ¯p, ¯v∗)(v∗∗)
=
(
Ω1(¯x, ¯p, ¯v∗)(v∗∗), for hv∗∗, ∇xψ(¯x, ¯p)i = 0
∅, for hv∗∗, ∇xψ(¯x, ¯p)i 6= 0,
(4.7)
where Ω1(¯x, ¯p, ¯v∗)(v∗∗) is defined by (4.3)
(iii) If ψ(¯x, ¯p) = 0, ∇xψ(¯x, ¯p) 6= 0, ∇pψ(¯x, ¯p) 6= 0, and ¯v∗ = 0, then for every v∗∗ ∈ X∗∗
one has
b
D∗F (¯x, ¯p, ¯v∗)(v∗∗) =
(
Ω2(¯x, ¯p, ¯v∗)(v∗∗), for hv∗∗, ∇xψ(¯x, ¯p)i ≥ 0
∅, for hv∗∗, ∇xψ(¯x, ¯p)i < 0, (4.8) and
D∗F (¯x, ¯p, ¯v∗)(v∗∗) =
{(0X∗, 0P∗)}, for hv∗∗, ∇xψ(¯x, ¯p)i < 0
Ω2(¯x, ¯p, ¯v∗)(v∗∗), for hv∗∗, ∇xψ(¯x, ¯p)i > 0
Ω3(¯x, ¯p, ¯v∗)(v∗∗), for hv∗∗, ∇xψ(¯x, ¯p)i = 0,
(4.9)
where Ω2(¯x, ¯p, ¯v∗)(v∗∗) and Ω3(¯x, ¯p, ¯v∗)(v∗∗) are respectively given by (4.4) and (4.5)
The following lemma provides us with important information on the SNC property of the normal cone operator F (·)
Lemma 4.1 The normal cone operator F (·) may be not SNC in some infinite dimensional Asplund spaces
Proof For X = `2, we have X∗∗ = X∗ = X = `2 Let (¯x, ¯p, ¯v∗) ∈ gphF We consider two cases of the point (¯x, ¯p, ¯v∗) below:
In the case ψ(¯x, ¯p) < 0: F (·) is not SNC at the point (¯x, ¯p, ¯v∗) Indeed, for every k ∈ IN , choose (xk, pk, vk∗, x∗k, p∗k, vk∗∗) ∈ gphF × X∗× P∗× X∗∗ as follows
(xk, pk, vk∗) = (¯x, ¯p, ¯v∗), (x∗k, p∗k) = (0, 0), v∗∗k = (0, , 0, 1, 0, )
Then, we have (x∗k, p∗k) ∈ bD∗F (xk, pk, vk∗)(v∗∗k ) for all k ∈ IN Letting k → ∞, we also have
(xk, pk, vk∗) → (¯x, ¯p, ¯v∗) and (x∗k, p∗k, v∗∗k )→ (0, 0, 0).w∗ However, when k → ∞, we obtain
k(x∗k, p∗k, v∗∗k )k = kx∗kk + kp∗kk + kvk∗k = 1 6→ 0
Therefore, F (·) is not SNC at (¯x, ¯p, ¯v∗)
In the case ψ(¯x, ¯p) = 0, ∇xψ(¯x, ¯p) 6= 0, ∇pψ(¯x, ¯p) 6= 0, and ¯v∗ = 0: F (·) is also not SNC at (¯x, ¯p, ¯v∗) Indeed, let (xk, pk, v∗k) = (¯x, ¯p, ¯v∗), (x∗k, p∗k) = (0, 0), and let vk∗∗= (0, , 0, 1, 0, )
or vk∗∗ = (0, , 0, −1, 0, ) such that hvk∗∗, ∇xψ(¯x, ¯p)i ≥ 0 for every k ∈ IN Then it holds that (xk, pk, vk∗) → (¯x, ¯p, ¯v∗) and (x∗k, p∗k, v∗∗k )→ (0, 0, 0) as k → ∞, andw∗
(x∗k, p∗k) ∈ bD∗F (xk, pk, vk∗)(v∗∗k ), ∀k ∈ IN
The last inclusion is due to Theorem 4.1(iii) Observe that k(x∗k, p∗k, vk∗∗)k = 1 6→ 0 as k → ∞
We have shown that in this case F (·) is also not SNC at (¯x, ¯p, ¯v∗) 2
...3 Coderivatives of implicit multifunctions< /h3>
In this section we establish formulas for computing/estimating the Fr´echet and Mordukhovich coderivatives of implicit multifunctions. .. the implicit multifunction S1(x, q) = {y ∈ Y | ∈ eQ(x, y, q)} we obtain all the desired
4 Solution stability of variational systems< /h3>
Lipschitzian stability of. .. Fr´echet and Mordukhovich coderivatives
of S(·) defined by (3.1) in the forthcoming theorem
Theorem 3.3 Consider the implicit multifunction (3.1) and let ¯ω := (¯x, ¯y) ∈ gphS and (¯x,