Computation of Graphical Derivative for a Class of
Normal Cone Mappings under a Very Weak Condition
Nguyen Huy Chieu∗ and
Le Van Hien
†
Abstract. Let Γ := {x ∈ Rn | q(x) ∈ Θ}, where q : Rn → Rm is a twice continuously differentiable mapping, and Θ is a nonempty polyhedral convex set in Rm . In this paper, we
first establish a formula for exactly computing the graphical derivative of the normal cone
mapping NΓ : Rn ⇒ Rn , x 7→ NΓ (x), under the condition that Mq (x) := q(x) − Θ is metrically subregular at the reference point. Then, based on this formula, we exhibit formulae for
computing the graphical derivative of solution mappings and present characterizations of the
isolated calmness for a broad class of generalized equations. Finally, applying to optimization,
we get a new result on the isolated calmness of stationary point mappings.
Key Words. Computation, Graphical Derivative, Normal Cone Mapping, Generalized Equation, Isolated Calmness
1
Introduction
Graphical derivative of a set-valued mapping at a point in its graph is the set-valued mapping
whose graph is the (Bouligand-Severi) tangent/contingent cone to the graph of the given set-valued
mapping at the reference point. This concept also known as the outer graphical derivative [24]
was introduced by Aubin [1] who called it the contingent derivative. The terminology “graphical
derivative” was used by Rockafellar and Wets [30], Dontchev and Rockafellar [10], and many other
people.
The graphical derivative is a powerful tool in variational analysis and its applications [2, 10, 30].
One can use it to investigate the stability and sensitivity of constraint and variational systems,
and more general, generalized equations [2, 10, 20, 21, 23, 24, 30], or even, characterize some
nice properties of set-valued mappings, such as the metric regularity, the Aubin/Lipschitz-like
property [2, 10, 11], the isolated calmness/the local upper Lipschitz property, the strong metric
subregularity [10, 21, 23]. Also, its graph can play a mediate role in computing the dual derivativelike constructions [8, 15, 19]. Although it is the key in dealing with some important issues in
variational analysis and its applications, unfortunately, computation of the graphical derivative
of a set-valued mapping is generally a challenging task. The problem has studied by many
researchers for a long time, and many interesting results in the direction have been established;
see [2, 10, 17, 19, 21, 23, 24, 30, 31] and the references therein.
In this paper, we concern the computation and application of the graphical derivative DNΓ of
the normal cone mapping NΓ : Rn ⇒ Rn , x 7→ NΓ (x), where Γ := {x | q(x) ∈ Θ} with q : Rn → Rm
∗
Department of Mathematics, Vinh University, Vinh, Nghe An, Vietnam; email: [email protected]. Research of this author was supported by the Vietnam National Foundation for Science and Technology Development
(NAFOSTED) under grant 101.01-2014.56.
†
Department of Natural Science Teachers, Ha Tinh University, Ha Tinh city, Ha Tinh, Vietnam; email:
[email protected].
1
being a twice continuously differentiable mapping, and Θ being a nonempty polyhedral convex
set in Rm . Note that normal cones keep certain first-order information on the underlying sets,
the derivative as well as the coderivative of the normal cone mappings are therefore objects of
second-order analysis. To our best knowledge, the first result closely related to the research
conducted in this paper was due to Dontchev and Rockafellar [8], where the authors gave an
exact description of the graph of DNΓ under the condition that Γ is a polyhedral convex set, and
then used it to compute the generalized Hessian/the limiting second-order subdifferential [26] of
the indicator function of Γ. The latter was a crucial step in characterizing the strong regularity for
variational inequalities over polyhedral convex sets in [8]. Henrion et al. [17, Remark 3.1] indicated
that a formula for computing the DNΓ can be obtained by [30, Corollary 13.43 & Exercise 13.17],
provided Mq (x) := q(x)−Θ is metrically regular around the reference point. Moreover, if Θ := Rm
−
and both the Magasarian-Fromovitz and the constant rank constraint qualifications are satisfied,
then this formula can be much more simplified [17, Theorems 3.1 & 3.2]. Gfrerer and Outrata [14,
Theorem 1] proved the formula holds if Θ := R`− and the metric regularity is replaced by the metric
subregularity at the reference point plus a uniform metric regularity around this point. Recently, in
1
the case Θ := {0Rm1 }×Rm−m
and under the metric subregularity, Gfrerer and Mordukhovich [15,
−
Theorem 5.1] showed that the result is still valid if the local uniform metric regularity is relaxed
to the so-called bounded extreme point property. For more information on recent results in this
direction as well as related issues, we refer the reader to [10, 13, 14, 15, 17, 19, 20, 27, 28, 29]
and the references therein. Among other things, the motivation of these studies came from the
applications to investigating isolated calmness for generalized equations [14, 17] and tilt stability
in optimization [15].
The aim of this paper is to extend and unify the above mentioned results on computation of
the graphical derivative DNΓ , and thereby obtain some new results on applications of the graphical derivative. More precisely, we first establish a formula for exactly computing the graphical
derivative of the normal cone mapping NΓ : Rn ⇒ Rn , x 7→ NΓ (x), under the condition that
Mq (x) := q(x) − Θ is metrically subregular at the reference point. Then, based on this formula,
we exhibit formulae for computing the graphical derivative of solution mappings and present characterizations of the isolated calmness for a broad class of generalized equations. Finally, applying
to optimization, we get a new result on the isolated calmness of stationary point mappings.
The structure of the paper is as follows. After recalling some basic notions and needed properties from variational analysis in the next section, we present the formula for computing the
graphical derivative of the normal cone mappings in Section 3. Then, in Section 4, we show how
to use this formula to compute the graphical derivative of solution mappings as well as derive
the new results on the isolated calmness for generalized equations and stationary point mappings.
Finally, we conclude the paper in Section 5 where we give some remarks on the perspective of the
obtained results.
2
Preliminaries
In this paper, all spaces are assumed to be Euclidean spaces with scalar product h·, ·i and Euclidean
norm k · k. For a given set C ⊂ Rn , the distance from x ∈ Rn to C is denoted by dC (x), that is,
k
P
dC (x) := inf kx − uk | u ∈ C , and posC :=
λi ci | λi ≥ 0, ci ∈ C ∪ {0}, i = 1, . . . k, k ∈ N ,
i=1
C ⊥ := u ∈ Rn | hu, xi = 0 for all x ∈ C , and C 0 := u ∈ Rn | hu, xi ≤ 0 for all x ∈ C . The
closed ball with centre x̄ and radius r > 0 is denoted by Br (x̄), and put B := B1 (0). As usual, we
use the little-o notation as a Landau symbol, that is, for x → x̄, one has α = o(β) if and only if
lim α(x) = 0, lim β(x) = 0, and lim α(x)
β(x) = 0. For α ∈ R, one puts [α]+ := max{α, 0}.
x→x̄
x→x̄
x→x̄
2
Below are basic notions and facts from variational analysis, which are frequently used in the
sequel; see [26, 30] for more details.
Definition 2.1. ([30, Chapter 6]). Let Ω be a nonempty subset of Rn .
(i) The (Bouligand-Severi) tangent/contingent cone to the set Ω at x̄ ∈ Ω is defined by
TΩ (x̄) := v ∈ Rn | there exist tk ↓ 0, vk → v with x̄ + tk vk ∈ Ω for all k ∈ N .
bC (x̄) given by
(ii) The (Fréchet) regular normal cone to Ω at x̄ ∈ Ω is the set N
bΩ (x̄) := v ∈ Rn | lim sup hv, x − x̄i ≤ 0 ,
N
kx − x̄k
Ω
x→x̄
Ω
where x → x̄ means that x → x̄ with x ∈ Ω.
(iii) The (Mordukhovich) limiting/basic normal cone to Ω at x̄ ∈ Ω is the set NΩ (x̄) defined
by
bΩ (xk ) with vk → v .
NΩ (x̄) = v ∈ Rn | there exist xk → x̄, vk ∈ N
bΩ (x̄) := ∅ by convention.
If x̄ 6∈ Ω, put NΩ (x̄) = N
It is known [26, Theorem 1.10] that the regular normal cone is the dual of the tangent cone:
bΩ (x̄) = {v ∈ Rn | hv, ui ≤ 0 for all u ∈ TΩ (x̄) .
N
(2.1)
If Ω is convex, the above tangent cone and normal cones reduce to the tangent cone and normal
cone in the sense of convex analysis.
n
m
Definition
mapping with its graph
2.2. (see [26, 30]). Let Φ : R ⇒ R be a set-valued
gphΦ := (x, y) | y ∈ Φ(x) and its domain DomΦ := x | Φ(x) 6= ∅ .
(i) Given a point x̄ ∈ DomΦ, the graphical derivative of Φ at x̄ for ȳ ∈ Φ(x̄) is the set-valued
mapping DΦ(x̄|ȳ) : Rn ⇒ Rm defined by
DΦ(x̄|ȳ)(v) := w ∈ Rm | (v, w) ∈ TgphΦ (x̄, ȳ) for all v ∈ Rn ,
that is, gphDΦ(x̄|ȳ) := TgphΦ (x̄, ȳ).
(ii) The regular coderivative of Φ at a given point (x̄, ȳ) ∈ Rn × Rm is the set-valued mapping
∗
b
D Φ(x̄, ȳ) : Rm ⇒ Rn defined by
b ∗ Φ(x̄, ȳ)(y ∗ ) := x∗ ∈ Rn | (x∗ , −y ∗ ) ∈ N
bgphΦ (x̄, ȳ) for all y ∗ ∈ Rm .
D
b ∗ Φ(x̄) for DΦ(x̄|ȳ) and D
b ∗ Φ(x̄, ȳ), respectively.
In the case Φ(x̄) = {ȳ}, one writes DΦ(x̄) and D
We note that if Φ : Rn → Rm is a single-valued mapping that is differentiable at x̄, then
b ∗ Φ(x̄) = ∇Φ(x̄)∗ .
DΦ(x̄) = ∇Φ(x̄) and D
Definition 2.3. ([26, 30]). Let ϕ : Rn → R̄ and x̄ ∈ Rn with ȳ := ϕ(x̄) finite.
(i) The regular subdifferential (known also as the presubdifferential and as the Fréchet/viscosity
subdifferential) of ϕ at x̄ is defined by
b
bepiϕ (x̄, ȳ) ,
∂ϕ(x̄)
:= x∗ ∈ Rn | (x∗ , −1) ∈ N
where epiϕ := (x, α) ∈ Rn × R | α ≥ ϕ(x) is the epigraph of ϕ.
(ii) The limiting subdifferential (known also as the Mordukhovich/basic subdifferntial) of ϕ at
x̄ is defined by
∂ϕ(x̄) := x∗ ∈ Rn | (x∗ , −1) ∈ Nepiϕ (x̄, ȳ) .
b
If |ϕ(x̄)| = ∞ then put ∂ϕ(x̄) = ∂ϕ(x̄)
:= ∅ by convention.
3
b
b
Note that ∂ϕ(x̄)
⊂ ∂ϕ(x̄) and if ϕ is a convex function then both ∂ϕ(x̄)
and ∂ϕ(x̄) coincide
with the subdifferential in the sense of convex analysis:
b
∂ϕ(x̄)
= ∂ϕ(x̄) = x ∈ Rn | hx∗ , x − x̄i ≤ ϕ(x) − ϕ(x̄) for all x ∈ Rn .
b
In addition, it is not difficult to see that for x̄ ∈ Rn with ϕ(x̄) ∈ R, one has x∗ ∈ ∂ϕ(x̄)
if and
only if
ϕ(x) − ϕ(x̄) − hx∗ , x − x̄i
lim inf
≥ 0.
x→x̄
kx − x̄k
Recall [10, Section 3.8] that a set-valued mapping Φ : Rn ⇒ Rm is said to be metrically
subregular at x̄ for ȳ if (x̄, ȳ) ∈ gphΦ and there exist κ, r > 0 such that
d x, Φ−1 (ȳ) ≤ κd ȳ, Φ(x) for all x ∈ Br (x̄).
Using this property, Gfrerer and Mordukhovich [15] introduced the following constraint qualification in the nonlinear programming setting.
Definition 2.4. ([15, Definition 3.2]). Consider the constraint set
Γ := {x ∈ Rn | q(x) ∈ Θ},
where q : Rn → Rm is a continuously differentiable mapping and Θ is a nonempty closed set in
Rm . One says that the metric subregularity constraint qualification (MSCQ) holds at x̄ ∈ Γ if
Mq (x) := q(x) − Θ is metrically subregular at x̄ for 0.
In fact, if Γ is the constraint set of a nonlinear programming, then the validity of MSCQ
amounts to the existence of a local error bound [18], which is weaker than most known constraint
qualifications [3, 4, 5, 16, 22, 25]. Furthermore, from definition we see that MSCQ is fulfilled at
every x ∈ Γ near x̄ whenever it holds at x̄ ∈ Γ.
Combining [12, Proposition 3.4] and [26, Corollary 1.15], we are able to get the following
formula for computing the normal cones to inverse sets.
Lemma 2.5. Let q : Rn → Rm be a twice continuously differentiable mapping, and let Θ be a
nonempty closed convex set in Rm and x̄ ∈ Γ := {x | q(x) ∈ Θ}. Suppose that MSCQ holds at x̄.
Then, there exists δ > 0 such that
bΓ (x) = ∇q(x)∗ NΘ (y) for all x ∈ Γ ∩ Bδ (x̄) and y := q(x).
NΓ (x) = N
3
Graphical Derivative of Normal Cone Mapping
From now on, we assume that Γ := {x | q(x) ∈ Θ}, where q : Rn → Rm is a twice continuously
differentiable mapping and Θ := {y ∈ Rm | hbi , yi ≤ αi , i = 1, 2, ..., `}, with bi ∈ Rm \{0} and
αi ∈ R. Fix x̄ ∈ Γ and x̄∗ ∈ NΓ (x̄), and put Λ := {λ ∈ NΘ (ȳ) | ∇q(x̄)T λ = x̄∗ }, ȳ := q(x̄), and
Iq (x̄) := {i = 1, 2, . . . , ` | hbi , ȳi = αi }, and K := TΓ (x̄) ∩ {x̄∗ }⊥ .
Since Θ is a nonempty polyhedral convex set, by Lemma 2.5, if MSCQ holds at x̄, then Λ is
a nonempty polyhedral convex set. In this case, for each v ∈ K, the problem LP(v)
min −v T ∇2 λT q (x̄)v
subject to λ ∈ Λ
is a linear programming with its dual program DP(v)
max hx̄∗ , zi
subject to ∇q(x̄)z + v T ∇2 q(x̄)v ∈ TΘ (ȳ),
4
where TΘ (ȳ) can be computed by TΘ (ȳ) = {w ∈ Rm | hbi , wi ≤ 0 for all i ∈ Iq (x̄)}; see [6, p.126].
To proceed, we need the following auxiliary result.
Lemma 3.1. Suppose that MSCQ is valid at x̄ and ȳ := q(x̄). Then, for each v ∈ K and λ ∈ Λ,
one has
b (v) = ∇q(x̄)T µ | µT ∇q(x̄)v = 0, µ ∈ TN (ȳ) (λ) ,
N
(3.1)
K
Θ
where NΘ (ȳ) = pos{bi | i ∈ Iq (x̄) , Iq (x̄) := i = 1, 2, . . . , ` | hbi , ȳi = αi , and
TNΘ (ȳ) (λ) = pos{bi | i ∈ Iq (x̄) − R+ λ.
Consequently, for v ∈ K, one has


 X
 ⊥
T
∗
b (v) =
N
t
b
∇q(x̄)
−
t
x̄
|
t
,
t
∈
R
,
i
∈
I
(x̄)
∩ v .
i i
0
0 i
+
q
K


(3.2)
i∈Iq (x̄)
Proof. We first prove the following inclusion holds:
b (v) ⊂ ∇q(x̄)T µ | µT ∇q(x̄)v = 0, µ ∈ TN (ȳ) (λ) .
N
K
Θ
(3.3)
b (v). Since N
b (v) = K 0 ∩ {v}⊥ = ∇q(x̄)T NΘ (ȳ) + Rx̄∗ ∩ {v}⊥ , there exist
Take any v ∗ ∈ N
K
K
µ̃ ∈ NΘ (ȳ) and α ∈ R such that v ∗ = ∇q(x̄)T (µ̃) + αx̄∗ = ∇q(x̄)T (µ̃ + αλ). For µ := µ̃ + αλ,
noting that λ, µ̃ ∈ NΘ ȳ), we have λ + k −1 µ = (1 + k −1 α)λ + k −1 µ̃ ∈ NΘ (ȳ) for all k ∈ N. The
latter implies µ ∈ TNΘ (ȳ) (λ). This together with v ∗ = ∇q(x̄)T µ and µT ∇q(x̄)v = hv ∗ , vi = 0
shows that (3.3) holds. We next justify the converse inclusion. Take an arbitrary µ ∈ TNΘ (ȳ) (λ)
with µT ∇q(x̄)v = 0. Then, we have λ + tµ ∈ NΘ (ȳ) for some t > 0. So, by Lemma 2.5, it holds
∇q(x̄)T (λ + tµ) ∈ NΓ (x̄) = TΓ (x̄)0 . Hence,
0 ≥ h∇q(x̄)T (λ + tµ), wi = hx̄∗ , wi + tµT ∇q(x̄)w = tµT ∇q(x̄)w,
0
for all w ∈ K, which guarantees ∇q(x̄)T µ ∈ K . This together with µT ∇q(x̄)v = 0 implies
b (v). So, equality (3.1) has been justified. Finally, it is not difficult to see that (3.2)
∇q(x̄)T µ ∈ N
K
is a straightforward consequence of (3.1).
We now arrive at the main result of this section.
Theorem 3.2. Let MSCQ be satisfied at x̄ ∈ Γ, and x̄∗ ∈ NΓ (x̄). Then, one has
b (v) .
TgphNΓ (x̄, x̄∗ ) = (v, v ∗ ) ∈ Rn × Rn | ∃λ ∈ Λ(v) : v ∗ ∈ ∇2 λT q (x̄)v + N
K
(3.4)
In other words, the graphical derivative of the normal cone mapping x 7→ NΓ (x) is given by
b (v).
DNΓ (x̄|x̄∗ )(v) = ∇2 λT q (x̄)v | λ ∈ Λ(v) + N
(3.5)
K
b (v) can
Here Λ(v) is the optimal solution set of the linear programming LP(v), and the cone N
K
be computed by (3.2).
Proof. We first justify the inclusion
b (v) ⊂ TgphN (x̄, x̄∗ ).
(v, v ∗ ) ∈ Rn × Rn | ∃λ ∈ Λ(v) : v ∗ ∈ ∇2 λT q (x̄)v + N
Γ
K
(3.6)
b
Take any (v,
v∗ ) ∈ Rn × Rn with v ∗ ∈ ∇2 λT q (x̄)v
+ NK (v) for some λ ∈ Λ(v). Let α > 0 satisfy
2
T
−1
2
T
αk∇ λ q (x̄ k < 2 . Then 2I + ∇ (αλ) q (x̄ is positively definite. Since Θ is a polyhedral
5
convex set, by [6, Theorem 3.70], there exists r > 0 such that x̄ is the unique global solution of
the problem (P ):
min kx̄ + αx̄∗ − xk2 subject to x ∈ Γ ∩ Br (x̄).
Since λ is an optimal solution to LP (v), the optimal solution set to DP (v) is nonempty. Moreover,
by [6, Proposition 2.191], for any optimal solution z to DP (v), we have
λ, ∇q(x̄)z + v T ∇2 q(x̄)v = 0.
b (v), by Lemma 3.1, it follows that v ∗ = ∇2 λT q (x̄)v + ∇q(x̄)T µ,
From v ∗ ∈ ∇2 λT q (x̄)v + N
K
⊥
for some µ ∈ TNΘ (ȳ) (λ) ∩ ∇q(x̄)v , where ȳ := q(x̄). Thus
hv ∗ , vi = v T ∇2 λT q (x̄)v + µT ∇q(x̄)v = v T ∇2 λT q (x̄)v.
Since Θ is a polyhedral convex set, λ ∈ NΘ (ȳ), and µ ∈ TNΘ (ȳ) (λ), there exists t̄ > 0 satisfying
λ + tµ ∈ NΘ (ȳ) for all t ∈ (0, t̄). Furthermore, the t̄ can be chosen such that
1
q(x̄ + tv + t2 z), bi < αi for all i 6∈ Iq (x̄).
2
(3.7)
For each t ∈ (0, t̄), pick an arbitrary global solution xt to the problem (Pt ):
1
min kx̄ + tv + t2 z + α(x̄∗ + tv ∗ ) − xk2 subject to x ∈ Γ ∩ Br (x̄).
2
Then xt → x̄ as t ↓ 0. Indeed, if this is not true, there exist tk ↓ 0 and xtk ⊂ Γ ∩ Br (x̄)
converging to some x0 ∈ Γ ∩ Br (x̄) with x0 6= x̄. Since xtk is a global solution to (Ptk ), it holds
1
1
kx̄ + tk v + t2k z + α(x̄∗ + tk v ∗ ) − xk2 ≥ kx̄ + tk v + t2k z + α(x̄∗ + tk v ∗ ) − xtk k2 ,
2
2
for all x ∈ Γ ∩ Br (x̄). Letting k → ∞, we get
kx̄ + αx̄∗ − xk2 ≥ kx̄ + αx̄∗ − x0 k2 for all x ∈ Γ ∩ Br (x̄),
which shows that x0 is a global solution to (P ). This contradicts the global solution uniqueness
of the problem (P ). So lim xt = x̄ and hence we can consider xt ∈ intBr (x̄) for all t ∈ (0, t̄).
t↓0
According to the first order optimality condition [30, Theorem 10.1],
α(x̄∗ + tv ∗ ) + t
1
1 (x̄ + tv − xt ) + tz ∈ NΓ (xt ) for all t ∈ (0, t̄).
t
2
The latter implies
lim
t↓0
dNΓ (xt ) (x̄ + tv ∗ )
= 0,
t
(3.8)
provided
1
lim (x̄ + tv − xt ) = 0.
t↓0 t
(3.9)
Moreover, (3.8) and (3.9) together guarantee (v, v ∗ ) ∈ TgphNΓ (x̄, x̄∗ ). So, we next prove (3.9) holds.
Since v ∈ K and MSCQ is fulfilled at x̄, we have ∇q(x̄)v ∈ TΘ (ȳ), that is,
∇q(x̄)v, bi ≤ 0 for all i ∈ Iq (x̄).
(3.10)
6
On the other hand, since z is feasible for the dual problem DP (v),
h∇q(x̄)z, bi i + hv T ∇2 q(x̄)v, bi i ≤ 0 for all i ∈ Iq (x̄).
(3.11)
Thus, from (3.10) and (3.11) it follows that
hq(x̄ + tv + 21 t2 z), bi i = hq(x̄), bi i + th∇q(x̄)(v), bi i
+ 21 t2 h∇q(x̄)z, bi i + hv T ∇2 q(x̄)v, bi i + o(t2 )
(3.12)
≤ αi + o(t2 ),
for all i ∈ Iq (x̄) and t ∈ (0, t̄). Since Mq is metrically subregular at x̄ for 0, we can assume, for
some κ1 > 0,
1
1
dΓ (x̄ + tv + t2 z) ≤ κ1 dΘ q(x̄ + tv + t2 z) for all t ∈ (0, t̄).
2
2
m
Note that Θ := {y ∈ R | hbi , yi ≤ αi , i = 1, 2, ..., `}. By Hoffman’s lemma [6, Theorem 2.200],
there exists κ2 > 0 such that
dΘ
`
X
1
1 2 q(x̄ + tv + t z) ≤ κ2
bi , q(x̄ + tv + t2 z) − αi + for all t ∈ (0, t̄).
2
2
(3.13)
i=1
Combining (3.7) and (3.12)-(3.13), we get
1
d x̄ + tv + t2 z, Γ ≤ o(t2 ) for all t ∈ (0, t̄).
2
This implies that, for each t ∈ (0, t̄), there exists x̃t ∈ Γ such that
1
kx̄ + tv + t2 z − x̃t k = o(t2 ),
2
(3.14)
which ensures that lim x̃t = x̄. Choosing the t̄ smaller if necessary, we assume x̃t ∈ Γ ∩ Br (x̄) for
t↓0
all t ∈ (0, t̄). Since xt is a global solution to (Pt ), we have
1
1
kx̄ + tv + t2 z + α x̄∗ + tv ∗ − xt k2 ≤ kx̄ + tv + t2 z + α x̄∗ + tv ∗ − x̃t k2 ,
2
2
and consequently,
kx̄ + tv + 12 t2 z − xt k2 + 2αhx̄∗ + tv ∗ , x̄ + tv + 21 t2 z − xt i
≤ kx̄ + tv + 12 t2 z − x̃t k2 + 2αhx̄∗ + tv ∗ , x̄ + tv + 21 t2 z − x̃t i
≤ kx̄ + tv + 12 t2 z − x̃t k2 + 2αkx̄∗ + tv ∗ kkx̄ + tv + 12 t2 z − x̃t k.
Combining this with (3.14), we arrive at
1
1
kx̄ + tv + t2 z − xt k2 + 2αhx̄∗ + tv ∗ , x̄ + tv + t2 z − xt i ≤ o(t2 ).
2
2
7
(3.15)
Since hbi , q(xt ) − q(x̄)i ≤ 0 for all i ∈ Iq (x̄), it holds q(xt ) − q(x̄) ∈ TΘ (ȳ). So, taking into account
that λ + tµ ∈ NΘ (ȳ) and ∇q(x̄)T µ = v ∗ − ∇2 (λT q)(x̄)v, we have
0 ≥ λ + tµ, q(xt ) − q(x̄)
= λ + tµ, ∇q(x̄)(xt − x̄) + 21 (xt − x̄)T ∇2 q(x̄)(xt − x̄) + o(kxt − x̄k2 )
= ∇q(x̄)T λ + t∇q(x̄)T µ, xt − x̄
+ 21 (xt − x̄)T ∇2 λT q (x̄)(xt − x̄) + o(kxt − x̄k2 )
= hx̄∗ + tv ∗ , xt − x̄i − th∇2 (λT q)(x̄)v, xt − x̄i
+ 21 (xt − x̄)T ∇2 λT q (x̄)(xt − x̄) + o(kxt − x̄k2 ),
showing that
1
−hx̄∗ + tv ∗ , xt − x̄i ≥ −th∇2 (λT q)(x̄)v, xt − x̄i + (xt − x̄)T ∇2 λT q (x̄)(xt − x̄) + o(kxt − x̄k2 ).
2
Hence, using the fact that hλ, ∇q(x̄)z +v T ∇2 q(x̄)vi = 0, hv ∗ , vi = v T ∇2 (λT q)(x̄)v and hx̄∗ , vi = 0,
we get the estimate:
hx̄∗ + tv ∗ , x̄ + tv +
t2
2z
− xt i = −hx̄∗ + tv ∗ , xt − x̄i +
t2
∗
2 hx̄ , zi
+ t2 hv ∗ , vi + o(t2 )
≥ −th∇2 (λT q)(x̄)v, xt − x̄i + 21 (xt − x̄)T ∇2 λT q (x̄)(xt − x̄) + 12 t2 h∇q(x̄)T λ, zi
+t2 v T ∇2 (λT q)(x̄)v + o(t2 ) + o(kxt − x̄k2 )
= −th∇2 (λT q)(x̄)v, xt − x̄i + 21 (xt − x̄)T ∇2 λT q (x̄)(xt − x̄)
− 12 t2 v T ∇2 (λT q)(x̄)v + o(t2 ) + o(kxt − x̄k2 )
= − 21 (x̄ + tv − xt )T ∇2 (λT q)(x̄)(x̄ + tv − xt ) + o(t2 ) + o(kxt − x̄k2 ),
which implies that
αhx̄∗ + tv ∗ , x̄ + tv +
t2
2z
− xt i ≥ − 21 α(x̄ + tv − xt )T ∇2 (λT q)(x̄)(x̄ + tv − xt )
+o(t2 ) + o(kxt − x̄k2 )
≥ − 21 αk∇2 (λT q)(x̄)kkx̄ + tv − xt k2 + o(t2 ) + o(kxt − x̄k2 )
≥ − 41 kx̄ + tv − xt k2 + o(t2 ) + o(kxt − x̄k2 ).
The latter is due to αk∇2 λT q (x̄ k < 2−1 . Consequently, by (3.15), we have
1
2 kx̄
+ tv − xt k2 + 41 t4 kzk2 + t3 hz, vi + t2 hz, x̄ − xt i
= kx̄ + tv + 21 t2 z − xt k2 − 12 kx̄ + tv − xt k2 ≤ kx̄ + tv + 12 t2 z − xt k2
+2αhx̄∗ + tv ∗ , x̄ + tv +
t2
2z
− xt i + o(t2 ) + o(kxt − x̄k2 )
≤ o(t2 ) + o(kxt − x̄k2 ).
8
Hence, the following estimate holds
kx̄ + tv − xt k2 ≤ o(t2 ) + o(kxt − x̄k2 ).
(3.16)
This allows us to choose t̃ > 0 such that, for each t ∈ (0, t̃),
1
kx̄ + tv − xt k2 ≤ (t2 + kxt − x̄k2 ),
2
or equivalently,
kx̄ − xt k2 + 2t2 kvk2 + 4thv, x̄ − xt i ≤ t2 .
The latter guarantees the validity of the following estimate
kx̄ − xt k − 2tkvk ≤ kx̄ + 2tv − xt k
=
p
p
kx̄ − xt k2 + 2t2 kvk2 + 4thv, x̄ − xt i + 2t2 kvk2 ≤ t 1 + 2kvk2 ,
for all t ∈ (0, t̃). Thus, we have
kx̄ − xt k ≤ t(2kvk +
p
1 + 2kvk2 ) for all t ∈ (0, t̃).
Combining this with (3.16) yields
kx̄ + tv − xt k = o(t),
that is, (3.9) is valid. By (3.8) and (3.9), we get (v, v ∗ ) ∈ TgphNΓ (x̄, x̄∗ ), and hence inclusion (3.6)
has been justified.
We next justify the converse inclusion:
b (v) .
TgphNΓ (x̄, x̄∗ ) ⊂ (v, v ∗ ) ∈ Rn × Rn | ∃λ ∈ Λ(v) : v ∗ ∈ ∇2 λT q (x̄)v + N
(3.17)
K
Take any (v, v ∗ ) ∈ TgphNΓ (x̄, x̄∗ ). By definition, there exist tk ↓ 0, vk → v, and vk∗ → v ∗ such that
x∗k := x̄∗ + tk vk∗ ∈ NΓ (xk ) with xk := x̄ + tk vk . Let Iq (xk ) := {i = 1, 2, . . . , ` | hbi , q(xk )i = αi }.
Using a subsequence if necessary, we can assume Iq (xk ) = I˜ for all k, with some fixed index set
I˜ ⊂ Iq (x̄). So, it holds
(
˜
= αi if i ∈ I,
hbi , q(x̄)i + tk hbi , ∇q(x̄)vk i + o(tk ) = hbi , q(xk )i
˜
< αi if i ∈ Iq (x̄)\I,
for all k. Thus
(
˜
= 0 if i ∈ I,
tk hbi , ∇q(x̄)vk i + o(tk )
˜
< 0 if i ∈ Iq (x̄)\I,
for all k. Dividing by tk and then letting k → ∞, we get
(
˜
= 0 if i ∈ I,
hbi , ∇q(x̄)vi
˜
≤ 0 if i ∈ Iq (x̄)\I.
(3.18)
To proceed, we next prove that there exist δ > 0, κ > 0 such that for all x ∈ Γ ∩ Bδ (x̄) and
x∗ ∈ Rn , one has
Λ(x, x∗ ) ∩ κkx∗ kBRm 6= ∅ whenever Λ(x, x∗ ) 6= ∅.
(3.19)
9
Indeed, let δ > 0 be such that MSCQ holds at every x ∈ Γ ∩ Bδ (x̄) with some constant γ > 0.
Take any x∗ ∈ Rn such that Λ(x, x∗ ) 6= ∅. If x∗ = 0 then 0 ∈ Λ(x, x∗ ) ∩ κkx∗ kBRm for any
bΓ (x) = ∇q(x)T NΘ (y) with y := q(x), and
κ > 0. Suppose now that x∗ 6= 0. Since NΓ (x) = N
∗
∗
−1
∗
b Γ (x).
bΓ (x) ∩ BRn . By [26, Corollary 1.96], N
bΓ (x) ∩ BRn = ∂d
Λ(x, x ) 6= ∅, it holds kx k x ∈ N
Hence, one has
dΓ (u) − dΓ (x) − hkx∗ k−1 x∗ , u − xi
lim inf
≥ 0.
u→x
ku − xk
Note that dΓ (x) = dΘ q(x) = 0 and dΓ (u) ≤ γdΘ q(u) for all u near x. We have
dΘ q(u) − dΘ q(x) − h(γkx∗ k)−1 x∗ , u − xi
lim inf
≥ 0.
u→x
ku − xk
b Θ ◦ q)(x) ⊂ ∂(dΘ ◦ q)(x). So, it follows from [26, Corollary 3.43]
This means (γkx∗ k)−1 x∗ ∈ ∂(d
and [26, Corollary 1.96] that
(γkx∗ k)−1 x∗ ∈ ∂(dΘ ◦ q)(x) ⊂ ∇q(x)T ∂dΘ (y) = ∇q(x)T NΘ (y) ∩ BRm ,
that is, (γkx∗ k)−1 x∗ = ∇q(x)T λ̃ for some λ̃ ∈ NΘ (y) ∩ BRm . Put λ := γkx∗ kλ̃ and κ := γ. We
have λ ∈ Λ(x, x∗ ) ∩ κkx∗ kBRm . Thus (3.19) holds.
M SCQ is fulfilled at x̄, we have Λ(xk , x∗k ) 6= ∅
Since x∗k ∈ NΓ (xk ), (xk , x∗k ) → (x̄, x̄∗ ) and
for all k sufficiently large. Note that {kx∗k k is bounded. By (3.19), there exist κ, c1 > 0 and
λk ∈ Λ(xk , x∗k ) such that
kλk k ≤ κkx∗k k ≤ c1 ,
for all k sufficiently large. Hence, by [3, Lemma 1] and using a subsequence if necessary, we
can assume λk → λ̃ ∈ Λ and λk ∈ Θ̃ := pos{bi | i ∈ I0 }, where I0 ⊂ I˜ and {bi }i∈I0 is linearly
independent. For each x∗ ∈ Rn , put
X
ΨI0 (x∗ ) := λ ∈ Rm | ∇q(x̄)T λ = x∗ , λ =
λi bi , λi ≥ 0 .
i∈I0
Since λk → λ̃, λk ∈ Θ̃, and Θ̃ is closed, it holds λ̃ ∈ Θ̃. So, λ̃ ∈ ΨI0 (x̄∗ ) and thus ΨI0 (x̄∗ ) 6= ∅.
On the other hand, since Θ̃ is a polyhedral convex cone in Rm , there exist ai ∈ Rm , i = 1, 2, ..., p,
such that Θ̃ = {λ ∈ Rm | hai , λi ≤ 0, i = 1, 2, ..., p}. Thus, we have
ΨI0 (x∗ ) := λ ∈ Rm | ∇q(x̄)T λ = x∗ , hai , λi ≤ 0, i = 1, 2, ..., p .
By Hoffman’s lemma [6, Theorem 2.200], there exists β > 0 such that
dΨI0 (x̄∗ ) (λk ) ≤ β k∇q(x̄)T λk − x̄∗ k +
p
X
hai , λk i + = βk∇q(x̄)T λk − x̄∗ k,
(3.20)
i=1
for all k. The above equality is due to λk ∈ Θ̃ for all k. Since q ∈ C 2 , for some c2 > 0, it holds
k∇q(x̄) − ∇q(xk )k ≤ c2 kx̄ − xk k = c2 tk kvk k,
for all k sufficiently large. Hence, we get the estimate:
k∇q(x̄)T λk − x̄∗ k = ktk vk∗ + ∇q(x̄) − ∇q(xk )
T
λk k
≤ tk kvk∗ k + k∇q(x̄) − ∇q(xk )kkλk k
≤ tk (kvk∗ k + c1 c2 kvk k),
10
for all k sufficiently large. Combining this with (3.20) yields that
dΨI0 (x̄∗ ) (λk ) ≤ βtk (kvk∗ k + c1 c2 kvk k),
for all k sufficiently large. Thus there exists λ̃k ∈ ΨI0 (x̄∗ ) such that
kλk − λ̃k k ≤ βtk (kvk∗ k + c1 c2 kvk k)
k
k
for all k sufficiently large. Put µk := λ t−kλ̃ . Due to the boundedness of {kµk k}, we can assume
P
P k
λi bi and
that {µk } converges to some µ ∈ Rm . Since λ̃, λk , λ̃k ∈ Θ̃, one has λ̃ =
λ̃i bi , λk =
i∈I0
i∈I0
P k
λ̃k =
λ̃i bi for some λ̃i , λki , λ̃ki ≥ 0. By (3.18), we get
i∈I0
(µk )T ∇q(x̄)v =
X 1
T
1 k
λ − λ̃k ∇q(x̄)v =
λki − λ̃ki bTi , ∇q(x̄)v = 0.
tk
tk
i∈I0
Letting k → ∞, we have
µT ∇q(x̄)v = 0.
(3.21)
Note that, by (3.18),
hx̄∗ , vi = h∇q(x̄)T λ̃k , vi =
X
λ̃ki bi , ∇q(x̄)v = 0.
i∈I0
Combining this with (3.18) allows us to conclude
v ∈ K.
(3.22)
Moreover, for each λ ∈ Λ, it holds
1
(λ̃k − λ)T q(xk ) = (λ̃k − λ)T q(x̄) + tk ∇q(x̄)vk + t2k (λ̃k − λ)T vkT ∇2 q(x̄)vk + o(t2k ).
2
P
λi bi for some λi ≥ 0. Thus,
Since λ ∈ NΘ (ȳ), we have λ =
(3.23)
i∈Iq (x̄)
(λ̃k − λ)T q(x̄) + tk ∇q(x̄)vk
= (λ̃k − λ)T q(x̄) + tk (λ̃k − λ)T ∇q(x̄)vk
= (λ̃k − λ)T q(x̄) =
P
(λ̃ki − λi )hbi , q(x̄)i
i∈Iq (x̄)
=
P
i∈I0
≤
P
i∈I0
(λ̃ki − λi )αi −
P
λ i αi
i∈Iq (x̄)\I0
(λ̃ki − λi )hbi , q(xk )i −
P
λi hbi , q(xk )i
i∈Iq (x̄)\I0
= (λ̃k − λ)T q(xk ).
This together with (3.23) guarantees 12 tk 2 (λ̃k − λ)T vkT ∇2 q(x̄)vk + o(t2k ) ≥ 0. Dividing the latter
by 21 tk 2 and then letting k → ∞, we have (λ̃ − λ)T v T ∇2 q(x̄)v ≥ 0 for all λ ∈ Λ, that is, λ̃ ∈ Λ(v).
11
Moreover, noting that ∇q(xk )T λk = x∗k = ∇q(x̄)T λ̃k + tk vk∗ , it holds
∇q(xk )T λk − ∇q(x̄)T λ̃k
k→∞
tk
v ∗ = lim vk∗ = lim
k→∞
= lim
∇q(xk ) − ∇q(x̄)
T
k→∞
= lim
k→∞
λ̃k + ∇q(xk )T (λk − λ̃k )
tk
(3.24)
T
tk ∇2 q(x̄)vk + o(tk ) λ̃k
∇q(xk )T (λk − λ̃k )
+ lim
k→∞
tk
tk
= ∇2 (λ̃T q)(x̄)v + ∇q(x̄)T µ.
Case 1: µ ∈ TNΘ (ȳ) (λ̃). By Lemma 3.1, from (3.21)-(3.22) and the fact λ̃ ∈ Λ it follows
b (v) with λ̃ ∈ Λ(v). This shows
b (v). Thus, by (3.24), v ∗ ∈ ∇2 (λ̃T q)(x̄)v + N
that ∇q(x̄)T µ ∈ N
K
K
∗
∗
(v, v ) ∈ T (x̄, x̄ ).
Case 2: µ 6∈ TNΘ (ȳ) (λ̃). Put
J := {i ∈ I0 | λ̃i = 0, µi < 0}.
We will show that J 6= ∅. Suppose to the contrary that J = ∅. Let
I˜0 := {i ∈ I0 | λ̃i = 0, µi = 0}
and
0
µ k :=
X
0
µik bi ,
i∈I0
where
(
µki if i ∈ I0 \I˜0
µi :=
0
if i ∈ I˜0
0k
and µki :=
λki − λ̃ki
.
tk
0
Taking into account the linear independence of {bi }i∈I0 , we get lim µ k = µ. Hence, by J = ∅,
k→∞
P
0
0
(λ̃i + tk µik )bi ∈ NΘ (ȳ), for all k sufficiently large. This implies µ ∈ TNΘ (ȳ) (λ̃),
λ̃ + tk µ k =
i∈I0
which is a contradiction. Thus J 6= ∅.
For each i ∈ J, since

µi < 0
λki −λ̃ki
tk
k→∞
 lim
there exists k̄ ∈ N such that
= µi ,
λk̄i − λ̃k̄i
µi
≤
< 0.
tk̄
2
Put
µ̃ := µ + 2
λ̃k̄ − λ̃ X
λ̃k̄ − λ̃i =
µi + 2 i
bi .
tk̄
tk̄
i∈I0
We next prove
µ̃ ∈ TNΘ (ȳ) (λ̃).
12
Note that if λ̃i = 0 then
µi + 2
λ̃k̄i − λ̃i
≥ µi ,
tk̄
and if i ∈ J then
µi + 2
λ̃k̄ − λ̃k̄i
µi
λ̃k̄i − λ̃i
≥ µi ≥ µi + 2 i
≥ µi − 2 = 0.
tk̄
tk̄
2
Thus, we have
λ̃k̄ − λ̃i
J 0 := i ∈ I0 | λ̃i = 0, µi + 2 i
< 0 = ∅.
tk̄
Put
µ̃k :=
λki − λ̃ki
λk̄ − λ̃i +2 i
bi ,
tk
tk̄
X
i∈I0 \J0
where
λ̃k̄ − λ̃i
J0 := i ∈ I0 | λ̃i = 0, µi + 2 i
=0 .
tk̄
Then lim µ̃k = µ̃, and
k→∞
λ̃ + tk µ̃k =
X
i∈J0
λ̃i bi +
X
λ̃i + λki − λ̃ki + 2
i∈I0 \J0
λk̄i − λ̃i tk bi .
tk̄
λk̄i −λ̃i
tk̄ tk > 0 for all k sufficiently
k
λ −λ̃k
lim i tk i = µi . We have
k→∞
Take any i ∈ I0 \J0 . If λ̃i > 0 then λ̃i + λki − λ̃ki + 2
then, by J 0 = ∅, µi + 2
λ̃k̄i −λ̃i
tk̄
> 0. Note that
λ̃i + λki − λ̃ki + 2
large. If λ̃i = 0
λk̄ − λ̃i λk̄i − λ̃i
λk − λ̃ki
+2 i
tk = λ̃i + tk i
> 0,
tk̄
tk
tk̄
for all k sufficiently large. Hence, λ̃ + tk µ̃k ∈ NΘ (ȳ) for all k sufficiently large. Due to the fact
lim µ̃k = µ̃, the latter guarantees µ̃ ∈ TNΘ (ȳ) (λ̃). In addition, it holds
k→∞
k̄
∇q(x̄)T µ̃ = ∇q(x̄)T µ + 2 λ̃ t−λ̃
k̄
= ∇q(x̄)T µ +
2
T
k̄
tk̄ ∇q(x̄) (λ̃
− λ̃)
= ∇q(x̄)T µ.
b (v) and thus v ∗ ∈ ∇2 (λ̃T q)(x̄)v + N
b (v) with λ̃ ∈ Λ(v).
By Lemma 3.1, we have ∇q(x̄)T µ̃ ∈ N
K
K
This shows that (3.17) holds and (3.4) has been justified.
Finally, we see that (3.4) is equivalent to (3.5), due to the definition of the graphical derivative.
The proof is complete.
Remark 3.3. In the above proof, we have combined the development of the approach and techniques of Gfrerer and Outrata [14] with some ideas from Ioffe and Outrata [12].
The following example exhibits the situation where the mapping Mq is metrically subregular,
but not metrically regular, while (3.4) and (3.5) hold.
13
Example 3.4. Let Θ := (y1 , y2 ) | − y1 ≤ 0, −y2 ≤ 0, y1 + y2 − 1 ≤ 0 , q(x) := (x2 + x21 , 0) for
all x = (x1 , x2 ) ∈ R2 , and x̄ := (0, 0, ). Then, one has
0 1
2
2
2
2
Γ := x ∈ R | q(x) ∈ Θ = x ∈ R | − x1 ≤ x2 ≤ 1 − x1 and ∇q(x̄) =
.
0 0
Furthermore, since (0, 0) 6∈ int q(x̄) + ∇q(x̄)R2 − Θ , by the Robinson-Ursescu theorem, Mq is not
metrically regular at x̄ for 0. However, Mq is metrically subregular at x̄ for 0, as dΓ (x) ≤ dΘ q(x)
for all x near x̄. Hence, Theorem 3.2 can be applicable for this situation. By a direct
verification,
2λ1 0
2
2
2
T
for all x ∈ R near x̄, it holds NΘ (0, 0) = R− , TΓ (0, 0) = R × R+ , ∇ λ q (x̄) =
, and
0 0


{(0, 0)}
if − x21 < x2


!

 2x
1
NΓ (x) =
R− if − x21 = x2

1



 ∅
otherwise.
Case 1: x̄∗ := (0, 0) ∈ NΓ (x̄). Then K = TΓ (0, 0) = R × R+ , Λ = {0} × R− , Λ(v) = {0} × R− ,
and
TgphNΓ (0, 0, 0, 0) = R × (0, +∞) × {(0, 0)} ∪ R × {(0, 0)} × R− .
Hence,
b (v)
(v, v ∗ ) ∈ R2 × R2 | ∃λ ∈ Λ(v) : v ∗ ∈ ∇2 λT q (x̄)v + N
K

2λ1 0
= (v, v ∗ ) ∈ R2 × R2 | ∃λ ∈ {0} × R− : v ∗ ∈ 
0

bR×R (v)
v + N
+
0
= R × (0, +∞) × {(0, 0)} ∪ R × {(0, 0)} × R−
= TgphNΓ (0, 0, 0, 0).
Case 2: x̄∗ := (0, −1) ∈ NΓ (x̄). Then K = TΓ (0, 0) ∩ {x̄∗ }⊥ = R × {0}, Λ = {−1} × R− ,
Λ(v) = {−1} × R− , and
TgphNΓ (0, 0, 0, −1) = (v1 , v2 , v1∗ , v2∗ ) ∈ R4 | v1∗ = −2v1 , v2 = 0}.
Hence,
b (v)
(v, v ∗ ) ∈ R2 × R2 | ∃λ ∈ Λ(v) : v ∗ ∈ ∇2 λT q (x̄)v + N
K
= (v, v ∗ ) ∈ R2 × R2 | ∃λ ∈ {−1} × R−


2λ1 0
bR×{0} (v)
v + N
: v∗ ∈ 
0 0
= (v1 , v2 , v1∗ , v2∗ ) ∈ R4 | v1∗ = −2v1 , v2 = 0}
= TgphNΓ (0, 0, 0, −1).
The above computation shows that (3.4) and (3.5) are valid.
14
For the case where Γ is a feasible set of a nonlinear programming, it may happen that the
assumption of Theorem 3.2 is fulfilled, while the bounded extreme point property in the sense of
[15, Definition 3.3] is invalid.
Example 3.5. Let q(x) := (−x1 , x1 − x21 x22 ), Θ := {(0, 0)}, Γ := x ∈ R2 | q(x) ∈ Θ = {0} × R,
and x̄ := (0, 0). We see that
−1 0
0 0
2
2 T
NΘ q(x̄) = R , TΓ (x̄) = {0} × R, ∇q(x̄) =
, ∇ (λ q)(x̄) =
, NΓ (x) = R × {0},
1 0
0 0
for all x ∈ Γ. Furthermore, since the gradients of the equality constraints at x̄ are linearly
dependent, the bounded extreme point property in the sense of [15, Definition 3.3] is invalid at x̄.
Obviously, Mq is metrically subregular at x̄ for 0, and so Theorem 3.2 can be applicable for this
situation. We next directly justify this
Let x̄∗ := (0, 0) ∈ NΓ (x̄). By simple computation,
claim.
−1
we have K = {0} × R, Λ(v) = Λ =
R, and TgphNΓ (0, 0, 0, 0) = R3 × {0}. Hence,
1
b (v)
(v, v ∗ ) ∈ R2 × R2 | ∃λ ∈ Λ(v) : v ∗ ∈ ∇2 λT q (x̄)v + N
K
b{0}×R (v)
= (v, v ∗ ) ∈ R2 × R2 | v ∗ ∈ N
= R3 × {0} = TgphNΓ (0, 0, 0, 0).
This shows that (3.4) and (3.5) hold.
The next result gives us a formula for computing the regular coderivative of the normal cone
mapping, which is a direct consequence of Theorem 3.2.
Corollary 3.6. Under the assumption of Theorem 3.2, one has
b ∗ NΓ (x̄, x̄∗ )(u∗ ) = u | hu, vi − u∗ , ∇2 λT q (x̄)v ≤ 0 for all v ∈ K, λ ∈ Λ(v), −u∗ ∈ T (v) .
D
K
b ∗ NΓ (x̄, x̄∗ )(u∗ ) if and only if (u, −u∗ ) ∈ N
bgphN (x̄, x̄∗ ).
Proof. By definition of coderivative, u ∈ D
Γ
Furthermore, by (2.1), the latter amounts to hu, vi − hu∗ , v ∗ i ≤ 0 for all (v, v ∗ ) ∈ TgphNΓ (x̄, x̄∗ ).
b ∗ NΓ (x̄, x̄∗ )(u∗ ) is
So, by Theorem 3.2, the necessary and sufficient condition for u ∈ D
b (v),
hu, vi − u∗ , ∇2 λT q (x̄)v + w ≤ 0 for all λ ∈ Λ(v) and w ∈ N
K
or equivalently,
b (v) 0 .
hu, vi − u∗ , ∇2 λT q (x̄)v ≤ 0 for all λ ∈ Λ(v), v ∈ K, −u∗ ∈ N
K
Hence, noting that K is convex, we arrive at the conclusion.
4
Application to Generalized Equation
In this section, we first consider the parametric generalized equation of the form:
0 ∈ F (x, y) + NΓ (x),
(4.25)
where F : Rn × Rs → Rn is a continuous differentiable mapping, x is a variable, y is a parameter,
and Γ := {x ∈ Rn | q(x) ∈ Θ}, Θ ⊂ Rm is a nonempty polyhedral set in Rm , and q : Rn → Rm is a
twice continuously differentiable mapping. Denote by S the solution mapping to (4.25) given by
S(y) := x ∈ Rn | 0 ∈ F (x, y) + NΓ (x) ,
15
which assigns the corresponding set of equilibria to each value of the parameter y.
Based on the result of the preceding section, we next provide a formula for computing the
graphical derivative of the solution mapping S.
Theorem 4.1. Let (ȳ, x̄) ∈ gphS and let Mq be metrically subregular at x̄ for 0. Then, one has
n
o
b (v) , (4.26)
DS(ȳ|x̄)(z) ⊂ v | − ∇y F (x̄, ȳ)z ∈ ∇x F (x̄, ȳ)v + ∇2 λT q (x̄)v | λ ∈ Λ(v) + N
K
for all z ∈ Rs . Inclusion (4.26) holds as equality if assume further that ∇y F (x̄, ȳ) is surjective.
Here K := TΓ (x̄) ∩ {x̄∗ }⊥ with x̄∗ := −F (x̄, ȳ), and Λ(v) is the optimal solution set of LP(v).
Proof. Let ϕ : Rs × Rn → Rn × Rn be the mapping given by
ϕ(y, x) := x, −F (x, y) for all (y, x) ∈ Rs × Rn .
Then gphS = ϕ−1 gphNΓ and ∇ϕ(ȳ, x̄)(z, v) = v, −∇x F (x̄, ȳ)v − ∇y F (x̄, ȳ)z for all (z, v).
Clearly, ∇y F (x̄, ȳ) is surjective if and only if ∇ϕ(ȳ, x̄) is surjective. So, by [30, Theorem 6.31],
we have
TgphS (ȳ, x̄) ⊂ (z, v) | v, −∇x F (x̄, ȳ)v − ∇y F (x̄, ȳ)z ∈ TgphNΓ (x̄, −F (x̄, ȳ)) ,
and the inclusion becomes equality if in addition ∇y F (x̄, ȳ) is surjective. By definition of graphical
derivative, we get the desired conclusion.
Remark 4.2. The surjectivity of ∇y F (x̄, ȳ) guarantees the parameterization (4.25) is ample at
x̄ in the sense of Dontchev and Rockafellar [10, p. 95]. This condition has been extensively used
in variational analysis [10, 26, 30].
If q is an affine mapping, then {∇2 λT q (x̄)v | λ ∈ Λ(v)} = {0} and Mq is automatically
metrically subregular. Hence, in this case, formula (4.26) can be much more simplified.
Corollary 4.3. ([9, Theorem 7.1]). Consider the generalized equation (4.25) with q : Rn → Rm
being an affine mapping. For any (ȳ, x̄) ∈ gphS and x̄∗ := −F (x̄, ȳ), one has
o
n
b (v) for all z ∈ Rs .
(4.27)
DS(ȳ|x̄)(z) ⊂ v | − ∇y F (x̄, ȳ)z ∈ ∇x F (x̄, ȳ)v + N
K
Inclusion (4.27) holds as equality if in addition ∇y F (x̄, ȳ) is surjective.
If Γ is the constraint set of a nonlinear programming satisfying the constant rank constraint
qualification (CRCQ) (see, e.g., [15]) at the reference point, then (4.26)
can be also simplified.
T
2
T
This is due to the facts that under CRCQ the mapping λ 7→ v ∇ λ q (x̄)v is constant on Λ for
b (v) for all λ1 , λ2 ∈ Λ and v ∈ K;
every v ∈ K, and further ∇2 λT1 q (x̄)v − ∇2 λT2 q (x̄)v ∈ N
K
see [14, Lemma 4], [15, Proposition 5.3], and [17, Corollary 3.2].
1
Corollary 4.4. Consider (4.25) with Θ := {0Rm1 } × Rm−m
and (ȳ, x̄) ∈ gphS. Assume that
−
CRCQ is fulfilled at x̄. Then, one has
n
o
b (v) ,
DS(ȳ|x̄)(z) ⊂ v | − ∇y F (x̄, ȳ)z ∈ ∇x F (x̄, ȳ)v + ∇2 λT q (x̄)v + N
(4.28)
K
for all z ∈ Rs and λ ∈ Λ. Inclusion (4.28) holds as equality if in addition ∇y F (x̄, ȳ) is surjective.
16
Proof. Suppose that CRCQ is fulfilled at x̄. Then Mq is metrically subregular at x̄ for 0. So, by
[15, Proposition 4.3], for each λ ∈ Γ, we have
(
(
)
+ (λ)
=
0
if
i
∈
E
∪
I
K = v ∈ Rn | h∇qi (x̄), vi
,
(4.29)
≤ 0 if i ∈ I(x̄)\I + (λ)
where E := {1, . . . , m1 }, I(x̄) := {i = m1 + 1, . . . , m | qi (x̄) = 0}, and I + (λ) := {i ∈ I(x̄) | λi > 0}.
On the other hand, by [15, Proposition 5.3 (i)], for each v ∈ Rn satisfying
[
h∇qi (x̄), vi = 0 whenever i ∈ E ∪ I + with I + :=
I + (λ)
λ∈Λ
the mapping λ 7→ v T ∇2 λT q (x̄)v is constant on Λ. Combining this with (4.29) shows that the
mapping λ 7→ v T ∇2 λT q (x̄)v is constant on Λ for each v ∈ K. Hence Λ(v) = Λ for all v ∈ K.
Next, following the proof scheme of [14, Lemma 4], we show that
b (v),
(4.30)
∇2 λT1 q (x̄)v − ∇2 λT2 q (x̄)v ∈ N
K
for all λ1 , λ2 ∈ Λ. Suppose to the contrary that there exist v ∈ K and λ1 , λ2 ∈ Λ such that
b (v).
∇2 λT1 q (x̄)v − ∇2 λT2 q (x̄)v 6∈ N
K
Thus, by the convex separation theorem, one can find some w satisfying the following condition
b (v).
hw, ∇2 λT1 q (x̄)v − ∇2 λT2 q (x̄)vi < hw, ui for all u ∈ N
K
From (3.2) it follows that

 X
b
NK (v) =

i∈E∪I(x̄)

 ⊥
∗
ti ∇qi (x̄) − t0 x̄ | ti ∈ R, i ∈ E, t0 , tj ≥ 0, j ∈ I(x̄) ∩ v .

Note that x̄∗ ∈ pos ∇qi (x̄) |, i ∈ I(x̄)
and ∇qi (x̄)v
= 0 for all i ∈ E ∪ I + . Hence, ∇qi (x̄)w = 0
for all i ∈ E ∪ I + and hw, ∇2 λT1 q (x̄)v − ∇2 λT2 q (x̄)vi < 0. By [15, Proposition 5.3 (ii)], there
exist λ̃ ∈ Λ with I + (λ̃) = I + and some ṽ ∈ K such that
(
= 0 if i ∈ E ∪ I + (λ̃)
h∇qi (x̄), ṽi
< 0 if i ∈ I(x̄)\I + (λ̃).
Consequently, hw + tṽ, ∇2 λT1 q (x̄)v − ∇2 λT2 q (x̄)vi < 0 and
(
= 0 if i ∈ E ∪ I + (λ̃)
h∇qi (x̄), w + tṽi
< 0 if i ∈ I(x̄)\I + (λ̃),
for all t > 0 sufficiently large. This allows
us to choose α, β > 0 such that w̃ := w + αṽ ∈ K,
v + β w̃ ∈ K, hw̃, ∇2 λT1 q (x̄)v − ∇2 λT2 q (x̄)vi < 0, and hence,
v + β w̃, ∇2 λT1 q (x̄)(v + β w̃) − ∇2 λT2 q (x̄)(v + β w̃) < 0.
This contradicts the fact that the mapping λ 7→ v T ∇2 λT q (x̄)v is constant on Λ for each v ∈ K.
So (4.30) holds. Noting that Λ(v) = Λ for all v ∈ K, by Theorem 4.1, the conclusion follows. 17
Next, we consider the so-called isolated calmness of S. This property introduced by Dontchev [7]
possesses a great of variationally attractive characteristics, such as stability with respect to approximation, having infinitesimal characterizations, or closely related to some other important
properties in variational analysis. For more information on the isolated calmness, we refer the
reader to the monograph [10].
Recall [10] that a given set-valued mapping Φ : Rs ⇒ Rn is said to be isolated calm at ȳ ∈ Rs
for x̄ ∈ Φ(ȳ) if there exist κ, r > 0 such that Φ(y) ∩ Br (x̄) ⊂ {x̄} + κky − ȳkBRn for all y ∈ Br (ȳ).
The isolated calmness is equivalent to the so-called strong metric subregularity of the inverse
[10, Theorem 3I.3]. It can be characterized by the graphical derivative. More concretely, we have
the following result whose necessity part was given in [21, Proposition 2.1] while the other part
was established in [23, Proposition 4.1].
Lemma 4.5. ([10, Theorem 4E.1]). Let Φ : Rs ⇒ Rn and let (ȳ, x̄) ∈ gphΦ. Then, Φ is isolated
calm at ȳ for x̄ if and only if DΦ(ȳ|x̄)(0) = {0}.
Now, combining Lemma 4.5 with Theorem 4.1, we are able to derive a characterization of the
isolated calmness.
Theorem 4.6. Let (ȳ, x̄) ∈ gphS and let Mq be metrically subregular at x̄ for 0. If the implication
b (v)
0 ∈ ∇x L(x̄, ȳ, λ)v + N
K
λ ∈ Λ(v), v ∈ Rn
⇒v=0
(4.31)
is valid, then S is isolated calm at ȳ for x̄. The inverse also holds if assume further that ∇y F (x̄, ȳ)
is surjective. Here L : Rn × Rs × Rm → Rn is given by L(x, y, λ) := F (x, y) + ∇q(x)T λ.
Proof. According to Theorem 4.1, we have
b (v)
DS(ȳ|x̄)(0) ⊂ v | ∃λ ∈ Λ(v) : 0 ∈ ∇x F (x̄, ȳ)v + ∇2 λT q (x̄)v + N
K
b (v) ,
= v | ∃λ ∈ Λ(v) : 0 ∈ ∇x L(x̄, ȳ, λ)v + N
K
and the inclusion becomes equality whenever ∇y F (x̄, ȳ) is surjective. Thus, by Lemma 4.5, we
get the desired conclusion.
Compared with [14, Theorem 7], we obtain here the same conclusion without the additional
assumption of the so-called uniformly metric regularity. Moreover, in the polyhedral convex case,
our result reduces to the one presented in the recent book of Dontchev and Rockafellar [10].
Corollary 4.7. ([10, Theorem 4G.1]). Consider the generalized equation (4.25) with Γ := Θ,
n = m, and q := In the identity mapping in Rn . Let (ȳ, x̄) ∈ gphS and x̄∗ := −F (x̄, ȳ). Then, if
∇x F (x̄, ȳ) + NK
−1
(0) = {0}
(4.32)
then S is isolated calm at ȳ for x̄. Moreover, if in addition rank∇y F (x̄, ȳ) = n, then property (4.32)
is necessary and sufficient for S to have the isolated calmness at ȳ for x̄.
Proof. Under the given assumption, it holds that the mapping Mq is metrically subregular at
b (v), Λ(v) = Λ(x̄, x̄∗ ) = {x̄∗ }, and ∇x L(x̄, ȳ, λ)v = ∇x F (x̄, ȳ)v. Thus (4.32)
x̄ for 0, NK (v) = N
K
is equivalent to (4.31). Moreover, rank∇y F (x̄, ȳ) = n if and only if ∇y F (x̄, ȳ) : Rs → Rn is
surjective. So, by Theorem 4.6, the conclusion follows.
We now consider the parametric generalized equation:
w ∈ F (x, y) + NΓ (x),
18
(4.33)
where F : Rn × Rs → Rn is a continuous differentiable mapping, x is the variable, and p := (y, w)
represents the parameter, and Γ := {x ∈ Rn | q(x) ∈ Θ} with Θ ⊂ Rm being a polyhedron and
q : Rn → Rm being a twice continuously differentiable mapping. Let S : Rs × Rn ⇒ Rn be the
solution mapping of (4.33), that is,
S(p) := x ∈ Rn | w ∈ F (x, y) + NΓ (x) for all p := (y, w) ∈ Rs × Rn .
(4.34)
The following result gives us a characterization of the isolated calmness of the mapping S(p).
Theorem 4.8. Let (p̄, x̄) ∈ gphS and let Mq be metrically subregular at (x̄, 0). Then, the following
assertions are equivalent:
(i) The implication
b (v)
0 ∈ ∇x L(x̄, p̄, λ)v + N
K
⇒v=0
λ ∈ Λ(v), v ∈ Rn
is valid.
(ii) The solution mapping S(p) is isolated calm at p̄ for x̄. Here L : Rn × Rs ×Rn ×Rm → Rn
is defined by L(x, p, λ) := F (x, y) − w + ∇q(x)T λ with p := (y, w).
Proof. For G(x, p) := F (x, y) − w, we have
S(p) = x ∈ Rn | 0 ∈ G(x, p) + NΓ (x) ,
and ∇Gp (x̄, p̄)(x, y, w) = ∇x F (x̄, ȳ)x + ∇y F (x̄, ȳ)y − w for all (x, y, w) ∈ Rn × Rs × Rn . The latter
shows that the partial derivative mapping ∇p G(x̄, p̄) : Rn × (Rs × Rn ) → Rn is surjective. Thus,
by Theorem 4.6, we get the desired conclusion.
Finally, we consider the parametric optimization problem:
minimize g(x, y) − hw, xi
subject to x ∈ Γ,
(4.35)
where g : Rn × Rs → R is twice continuously differentiable, Γ := {x ∈ Rn | q(x) ∈ Θ}, Θ is a
nonempty polyhedral convex set in Rm , q : Rn → Rm is twice continuously differentiable, x is a
variable, and y ∈ Rs and w ∈ Rn are parameters.
Recall that the set-valued mapping XKKT : Rs × Rn → Rn defined by
XKKT (p) := x ∈ Rn | 0 ∈ ∇x g(x, y) − w + NΓ (x) , p := (y, w) ∈ Rs × Rn ,
is called the stationary point mapping of (4.35).
Obviously, the stationary point mapping XKKT (p) is a special case of the set-valued mapping
S(p) given by (4.34). So, by Theorem 4.8, we get the corresponding characterization of isolated
calmness of the stationary point mapping of (4.35).
Corollary 4.9. Let (p̄, x̄) ∈ gphXKKT and let Mq be metrically subregular at (x̄, 0). Then, the
following assertions are equivalent:
(i) The implication
b (v)
0 ∈ ∇x L(x̄, p̄, λ)v + N
K
⇒v=0
λ ∈ Λ(v), v ∈ Rn
is valid.
(ii) The mapping XKKT (p) is isolated calm at p̄ for x̄. Here L : Rn × Rs × Rn × Rm → Rn
is defined by L(x, p, λ) := ∇x g(x, y) − w + ∇q(x)T λ with p := (y, w).
If in addition Γ is a convex set and g(x, y) is a convex function in x, then the stationary point
mapping XKKT (p) coincides with the optimal solution mapping of (4.35). Hence, in this case,
Corollary 4.9 shows us a characterization of isolated calmness for the optimal solution mapping
of the considered problem.
19
5
Concluding Remarks
The main results of this paper exhibit formulae for computing the graphical derivative of the
normal cone mappings and the solution mappings of generalized equations, under a very weak
condition. We also present characterizations of isolated calmness for a broad class of generalized
equations. This class covers the generalized equation reformulation of variational inequalities
over polyhedral convex sets, and especially, the stationary point mappings of parametric nonlinear
programmings. In our view, these results can be applied to the study of the stability and sensitivity
for more structural problems in which the specific features may help us to get better information
on the considered issues. In addition, we think that it would be very interesting if we could use the
obtained results on computation of graphical derivative to investigate the tilt-sability for general
optimization problems, and the metric regularity for the class of related set-valued mappings. We
will pay our attention to these topics in the coming time. Note that some nice results in this
direction were recently reported in the literature; see, e.g., [14, 15, 27].
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