Math. Program., Ser. A
DOI 10.1007/s10107-015-0877-2
FULL LENGTH PAPER
A nonsmooth Robinson’s inverse function theorem
in Banach spaces
R. Cibulka · A. L. Dontchev
Received: 24 January 2014 / Accepted: 17 February 2015
© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015
Abstract In a recent paper, Izmailov (Math Program Ser A 147:581–590, 2014)
derived an extension of Robinson’s implicit function theorem for nonsmooth generalized equations in finite dimensions, which reduces to Clarke’s inverse function
theorem when the generalized equation is just an equation. Páles (J Math Anal Appl
209:202–220, 1997) gave earlier a generalization of Clarke’s inverse function theorem
to Banach spaces by employing Ioffe’s strict pre-derivative. In this paper we generalize
both theorems of Izmailov and Páles to nonsmooth generalized equations in Banach
spaces.
Keywords Robinson’s inverse function theorem · Clarke’s inverse function
theorem · Strict pre-derivative · Generalized equation · Strong metric regularity
Mathematics Subject Classification
49J53 · 49J52 · 49K40 · 90C31
A. L. Dontchev: Supported by the National Science Foundation Grant DMS 1008341.
R. Cibulka
NTIS - New Technologies for the Information Society and Department of Mathematics,
Faculty of Applied Sciences, University of West Bohemia, Univerzitní 22, 306 14 Plzeň,
Czech Republic
e-mail: [email protected]
A. L. Dontchev (B)
Mathematical Reviews and Department of Mathematics, The University of Michigan,
Ann Arbor, MI, USA
e-mail: [email protected]
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R. Cibulka, A. L. Dontchev
1 Introduction
The classical inverse function theorem says that if a function f acting from a Banach
space X to itself is continuously Fréchet differentiable around a point x̄ ∈ int dom f
then the inverse f −1 has a single-valued graphical localization around f (x̄) for x̄
which is continuously differentiable around f (x̄) if and only if the Fréchet derivative
D f (x̄) is invertible, that is, D f (x̄)−1 is a linear function. Here and further we use
the terminology from [3] according to which, for a set-valued mapping F : X ⇒ Y
and ȳ ∈ F(x̄), a graphical localization of F at x̄ for ȳ is a mapping F̃ such that
gph F̃ = (U × V ) ∩ gph F for some neighborhoods U of x̄ and V of ȳ; clearly, dom F̃
depends on the choice of U and V . If a graphical localization at x̄ for ȳ of F is a
function whose domain is a neighborhood of x̄, we say that this localization is around
x̄ for ȳ. By employing his generalized Jacobian, Clarke [1] proved the following
nonsmooth inverse function theorem in finite dimensions: if a function f : Rn → Rn
is Lipschitz continuous around x̄, then the inverse f −1 has a Lipschitz continuous
single-valued graphical localization around f (x̄) for x̄ provided that every element of
the generalized Jacobian, denoted in this paper by ∂¯ f (x̄), is a nonsingular matrix. A
generalization of Clarke’s theorem was given by Kummer [8] who showed that, if a
function f : Rn → Rn is continuous around x̄, then f −1 has a Lipschitz continuous
single-valued graphical localization around f (x̄) for x̄ if and only if the so-called
strict graphical derivative of f at x̄ is injective. A different kind of nonsmooth inverse
function theorem was derived by Robinson [13] in which the approximation of the
underlying function is single-valued and has Lipschitz inverse.
Throughout X and Y are Banach spaces. The notation f : X → Y means that f
is a function while F : X ⇒ Y denotes a mapping whichmay be set-valued.
The
y ∈ F(x) , the domain
graph of a mapping F is the
set
gph
F
=
(x,
y)
∈
X
×
Y
of F is the set dom F = x ∈ X F(x) = ∅ , and the inverse of F is the mapping
y → F −1 (y) = x ∈ X y ∈ F(x) . The closed and open ball centered at x with
o
radius r are denoted by Br (x) and Br (x), respectively. A set C is said to be locally
closed at x̄ ∈ C when there exists r > 0 such that C ∩ Br (x̄) is closed. Whenever
a set is singleton we identify it with its only element, e.g., for the Minkowski sum
of a singleton {a} and a set C we write a + C instead of {a} + C. The space of all
linear bounded mappings acting from X to Y is equipped with the standard operator
norm and denoted by L(X, Y ). Given a subset A of L(X, Y ), recall that the measure
of non-compactness of A, denoted by χ (A), is defined by
Br (A) A ∈ B for some finite set B ⊂ A .
χ (A) = inf r > 0 A ⊂
Clearly, if A is compact then χ (A) = 0; conversely, if χ (A) = 0 then the closure of
A is compact.
In a seminal paper Robinson [12] extended the classical implicit function theorem
to generalized equations defined as inclusions of the form 0 ∈ f (x) + F(x) where f
is a function and F is a set-valued mapping. In the same paper Robinson coined the
concept of “strong regularity.” In the terminology of [3], given a set-valued mapping
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A nonsmooth Robinson’s inverse function theorem
F : X ⇒ Y with ȳ ∈ F(x̄), if F −1 has a Lipschitz continuous single-valued graphical
localization around ȳ for x̄, i.e., a graphical localization which is a Lipschitz continuous
function in a neighborhood of ȳ, then F is said to be strongly metrically regular at
x̄ for ȳ. Recall that the Lipschitz modulus of a function f : X → Y at x̄, denoted
by lip( f ; x̄), is the infimum of all l > 0 for which there exists a neighborhood U of
x̄ such that f is Lipschitz continuous on U with a Lipschitz constant l. Clearly, if a
mapping F is strongly metrically regular at x̄ for ȳ, then every graphical localization
of its inverse F −1 around ȳ for x̄ has the same Lipschitz modulus.
Robinson stated his theorem [12, Theorem 2.1] as an implicit function theorem in
linear normed spaces. The following symmetric inverse function version of Robinson’s
theorem fits our purposes:
Theorem 1 (Robinson’s inverse function theorem) Let f : X → Y be a function which
is strictly Fréchet differentiable at x̄ with strict Fréchet derivative D f (x̄) and let
F : X ⇒ Y be any set-valued mapping. Then the following are equivalent:
(a) the mapping f (x̄) + D f (x̄)(· − x̄) + F(·) is strongly metrically regular at x̄ for
ȳ;
(b) the mapping f + F is strongly metrically regular at x̄ for ȳ.
In either case, if h and g denote single-valued localizations of the inverses of the
mapping appearing in (a) and (b) respectively, then
lip(h; ȳ) = lip(g; ȳ).
Various versions of Robinson’s theorem, including Theorem 1 above, as well as
related results centered around metric regularity together with historical remarks, are
broadly discussed in the book [3].
In a recent paper A. F. Izmailov published the following result in [6, Theorem 1.3]
which extends both Clarke’s inverse function theorem and a finite-dimensional version
of Robinson’s theorem (the implication (a) ⇒ (b) in Theorem 1).
Theorem 2 ([6, Theorem 1.3]) Let f : Rn → Rn be a function which is Lipschitz
continuous around x̄, let ȳ ∈ Rn , and let F : Rn ⇒ Rn be a set-valued mapping.
Suppose that for every matrix J in ∂¯ f (x̄), the Clarke generalized Jacobian of f at x̄,
the mapping G J : x → f (x̄) + J (x − x̄) + F(x) is strongly metrically regular at x̄
for ȳ. Then the mapping f + F is strongly metrically regular at x̄ for ȳ.
When F is the zero mapping, then the strong regularity assumption in Theorem 2
reduces to the requirement every matrix in ∂¯ f (x̄) be nonsingular, and then Theorem 2
becomes Clarke’s inverse function theorem.
In this paper we present the following extension of Izmailov’s theorem to Banach
spaces:
Theorem 3 (Nonsmooth Robinson’s inverse function theorem in Banach spaces) Let
(x̄, ȳ) ∈ X × Y , let f : X → Y and F : X ⇒ Y be such that ȳ ∈ f (x̄) + F(x̄).
Suppose that there exist a convex subset A of L(X, Y ) and a constant c > 0 such that
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R. Cibulka, A. L. Dontchev
(A) there exists r > 0 such that for each u and v in Br (x̄) one can find A ∈ A such
that
f (v) − f (u) − A(v − u)
≤ c
v − u
;
(B) for every A ∈ A the mapping
G A : x → f (x̄) + A(x − x̄) + F(x)
is strongly metrically regular at x̄ for ȳ; moreover, if s A is any single-valued
graphical localization of G −1
A around ȳ for x̄, then
(c + χ (A)) sup lip(s A ; ȳ) < 1.
A∈A
Then the mapping f + F is strongly metrically regular at x̄ for ȳ.
A proof of Theorem 3 will be given in the following section. Here we first discuss the
assumptions (A) and (B) in its statement. Ioffe [4] introduced the strict pre-derivative
of f at x̄ which is a homogeneous set-valued mapping T : X ⇒ Y such that for each
ε > 0 there exists r > 0 such that
f (x1 ) ∈ f (x2 ) + T (x1 − x2 ) + ε
x1 − x2 B, whenever x1 , x2 ∈ Br (x̄).
When X = Y = Rn , as shown in [4, Corollary 9.11], the Clarke generalized Jacobian
∂¯ f (x̄) of a locally Lipschitz-continuous function f is a strict pre-derivative of f at
x̄ and hence, it satisfies assumption (A) with any positive c. In this case assumption
(B) holds exactly when for each J ∈ ∂¯ f (x̄) the mapping G J in Theorem 2 is strongly
metrically regular at x̄ for ȳ, that is, we obtain Izmailov’s theorem [6, Theorem 1.3].
The strict pre-derivative of f at x̄ is single-valued if and only if f is strictly differentiable at x̄; hence in this case we obtain Theorem 1. For a statement of Robinson’s
theorem in general metric spaces, see [3, Theorem 5F.1].
In a different direction, Páles extended in [10, Theorem 5] Clarke’s inverse function
theorem to functions acting in Banach spaces. Specifically, based on his preceding
paper [9], Páles assumed in [10] that the function f has a strict pre-derivative at x̄
generated by a convex set A whose measure of non-compactness is strictly less than
the infimum of the surjection moduli of the elements of A. This condition becomes
equivalent to our condition (B) when F is the zero mapping. Thus, our Theorem 3
extends [10, Theorem 5] to generalized equations.
Theorem 3 can be easily extended to an equivalent implicit function theorem for
the generalized equation 0 ∈ f ( p, x) + F(x), where P, X and Y are Banach spaces
and f : P × X → Y is Lipschitz continuous around a point ( p̄, x̄) such that 0 ∈
f ( p̄, x̄) + F(x̄). For such a generalization one may adapt to infinite dimensions the
way Clarke’s implicit function theorem is derived in the book of Clarke [2] from its
inverse function counterpart. In that case one needs to use a set A in L(P × X, Y )
whose projection on the space L(X, Y ) has properties analogous to those displayed
in assumptions (A), (B).
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A nonsmooth Robinson’s inverse function theorem
The authors are aware of the fact that assumptions (A) and (B) in Theorem 3 above
may be difficult to check in the general case considered. However, we expect that in
special cases, e.g., for optimal control problems, these assumptions can be reduced to
numerically tractable conditions. This is a subject of our continuing research on this
topic.
2 Proof of Theorem 3
We start with the following simple lemma:
Lemma 1 (Local graph closedness from strong metric regularity) Let f : X → Y be
a function which is Lipschitz continuous around x̄, let F : X ⇒ Y be a set-valued
mapping, and suppose that f + F is strongly metrically regular at x̄ for ȳ. Then gph F
is locally closed at (x̄, ȳ − f (x̄)).
Proof The assumption that f + F is strongly metrically regular at x̄ for ȳ means
that there exist positive α and β such that the mapping Bβ ( ȳ) y → s(y) :=
( f + F)−1 (y) ∩ Bα (x̄) is a Lipschitz continuous function. Let a > 0 and b > 0 be
such that f is Lipschitz continuous in Ba (x̄) with constant l, and moreover b +la ≤ β
and a ≤ α. Then dom s ⊃ Bb+la ( ȳ). Let (xn , z n ) ∈ gph F ∩ (Ba (x̄) × Bb ( ȳ − f (x̄)))
converge to (x, z) as n → ∞. For sufficiently large n we have
z n + f (xn ) − ȳ
≤ z n − ( ȳ − f (x̄))
+ f (xn ) − f (x̄)
≤ b + la.
Then z n + f (xn ) ∈ Bb+la ( ȳ) for large n and also z n + f (xn ) ∈ ( f + F)(xn );
hence, xn = s(z n + f (xn )). Since both f and s are continuous, passing to the limit
with n → ∞ we get x = s(z + f (x)) = ( f + F)−1 (z + f (x)) ∩ Ba (x̄), that is,
z + f (x) ∈ f (x) + F(x), hence (x, z) ∈ gph F. Thus, gph F is locally closed at
(x̄, ȳ − f (x̄)).
Without loss of generality, let ȳ = 0. The proof of Theorem 3 will include several
steps.
Step 1 Set m := sup A∈A lip(s A ; 0) with any choice of single-valued graphical localizations s A of G −1
A ; by assumption (B) this is a finite number. Choose ε and such
that
ε > c + χ (A), > m and ε < 1.
(1)
We will show that there is a compact convex subset B of a finite dimensional subspace
of L(X, Y ) such that
(a) for each two distinct u and v in Br (x̄) there is A ∈ B such that
f (v) − f (u) − A(v − u)
< ε
v − u
;
(b) sup A∈B lip(s A ; 0) < with any choice of single-valued graphical localizations
s A of G −1
A around 0 for x̄.
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R. Cibulka, A. L. Dontchev
To prove this, pick positive γ such that
γ m < 1, c + χ (A) + γ < ε, and m/(1 − γ m) < .
(2)
Since χ (A) < ∞, there is a finite set A F := {A1 , . . . , An } ⊂ A, with cardinality
n, say, such that A ⊂ A F + Bχ (A)+γ (0). Denote by B the closed convex hull of
A F . Clearly, B is a convex compact subset of the affine hull of AF (being a finite
dimensional subspace of L(X, Y )). Fix any two distinct elements u and v in Br (x̄).
From the assumption (A) there exists A0 ∈ A such that
f (v) − f (u) − A0 (v − u)
≤ c
v − u
.
Pick A ∈ A F ⊂ B such that
(A − A0 )h
≤ (χ (A) + γ )
h
for each h ∈ X.
Hence, the triangle inequality and the fact that u = v reveal that
f (v) − f (u) − A(v − u)
≤ (c + χ (A) + γ )
v − u
< ε
v − u
,
which is (a).
n
nFix an arbitrary A ∈ B. Hence, there is A := i=1 λi Ai for some λi ≥ 0 with
x̄).
+(A− A)(·−
i=1 λi = 1 such that A ∈ A and A− A
< γ . Note that G A = G A
−1
Let s A
be any single-valued graphical localization of G A
around 0 for x̄. Then by
assumption (B), lip(s A
; 0) ≤ m. Since γ m < 1 by (2), applying [3, Theorem 5F.1],
there is a single-valued graphical localization of G −1
A around 0 for x̄ with
lip(s A ; 0) ≤
m
< .
1 − γm
Since every graphical localization of this kind has the same Lipschitz modulus, (b) is
proved.
The next step shows that the strong metric regularity of G A is uniform in A ∈ B
in the sense that the neighborhoods and the Lipschitz constant associated with the
localization of G −1
A are the same for all A ∈ B.
Step 2 There exists β > 0 such that for every A ∈ B, the mapping Bβ (0) y →
G −1
A (y) ∩ Bβ ( x̄) is a Lipschitz continuous function with Lipschitz constant .
Fix positive κ and γ such that
2κγ < 1 and
sup lip(s A ; 0) < κ < (1 − κγ ).
A∈B
Choose any A ∈ B. The mapping G −1
A has a single-valued graphical localization
s A around 0 for x̄ which is Lipschitz continuous around 0 with the constant κ and
neighborhoods Bκb (x̄) and Bb (0) for some b > 0.
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A nonsmooth Robinson’s inverse function theorem
Choose any A ∈ B such that A − A < γ . Now consider the mapping G A
associated with A as in (B). For every y ∈ Bb/2 (0) and every x ∈ Bκb (x̄) we have
y + (A − A )(x − x̄)
< b/2 + γ κb < b/2 + b/2 = b.
Therefore, for every y ∈ Bb/2 (0) and every x ∈ Bκb (x̄),
x ∈ G −1
A (y) ⇐⇒ x ∈ ξ y (x) := s A (y + (A − A )(x − x̄)).
Let y ∈ Bb/2 (0). Then
ξ y (x̄) − x̄
= s A (y) − s A (0)
≤ κ
y
≤ κb/2.
Moreover, for any x, x ∈ Bκb (x̄) we obtain
ξ y (x) − ξ y (x )
≤ κ
(A − A )(x − x )
≤ κγ x − x .
Since κγ < 1/2, the above estimate with x := x̄ and the previous one imply that
ξ y (x) − x̄
≤ ξ y (x) − ξ y (x̄)
+ ξ y (x̄) − x̄
< κb/2 + κb/2 = κb
whenever x ∈ Bκb (x̄).
Therefore ξ y is a contraction from Bκb (x̄) into itself. By Banach’s contraction mapping
principle, there a unique fixed point x(y) of ξ y in Bκb (x̄). Since x(y) = G −1
A (y) ∩
Bκb (x̄) for every y ∈ Bb/2 (0), we conclude that the mapping Bb/2 (0) y → x(y) =
G −1
A (y) ∩ Bκb ( x̄) is single-valued. Furthermore, for any y, y ∈ Bb/2 (0) we have
x(y) − x(y )
= ξ y (x(y))−ξ y (x(y ))
≤ κ
y − y +κ
(A− A )(x(y)−x(y ))
≤ κ
y − y + κγ x(y) − x(y )
,
hence x(·) is Lipschitz continuous on Bb/2 (0) with Lipschitz constant κ/(1 − κγ ).
o
Thus, for any A ∈ B with A ∈ Bγ (A) the mapping G −1
A has a Lipschitz continuous single-valued localization around 0 for x̄ with a Lipschitz constant κ/(1 − κγ )
and neighborhoods Bκb (x̄) and Bb/2 (0). In other words, for any operator A which
is sufficiently close to A the sizes of the neighborhoods and the Lipschitz constant
associated with the localization of G −1
A remain the same.
Observe that κ and γ are independent of a particular A ∈ B which is not the case
o
for b. Pick any A ∈ B and the corresponding b to obtain that for every A ∈ Bγ (A) the
mapping Bb/2 (0) y → G −1
A (y) ∩ Bκb ( x̄) is a Lipschitz continuous function with
o
the constant κ/(1 − κγ ). Since B is compact, from the open covering ∪ A∈B Bγ (A) of
o
B we can choose a finite subcovering with open balls Bγ (Ai ); let bi be the constants
associated with the Lipschitz localizations for G −1
Ai . Taking β = min{1/2, κ/} mini bi
and observing that κ/(1 − κγ ) < Step 2 is complete.
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R. Cibulka, A. L. Dontchev
Step 3 Let μ := sup A∈B A
. Clearly, μ < ∞ since B is compact. Note that f is
Lipschitz continuous on Br (x̄) with a Lipschitz constant μ + ε. Indeed, for any two
distinct elements u and v of Br (x̄) condition (a) implies that, choosing an appropriate
A ∈ B, we get that
f (v) − f (u)
≤ f (v) − f (u) − A(v − u)
+ A(v − u)
< (ε + μ)
v − u
.
The combination of Lemma 1 and Step 2 then yields that there is a positive constant
β such that the following statements hold:
• the set gph F ∩ Bβ (x̄) × Bβ (− f (x̄)) is closed;
• f is Lipschitz continuous on Bβ (x̄) with the constant μ + ε;
• for each A ∈ B, the mapping
B2β (0) z → s A (z) := G −1
A (z) ∩ B2β ( x̄)
is a Lipschitz continuous function with Lipschitz constant .
Recall that satisfies (1). Define the mapping
B × Bβ (0) (A, z) → ϕ(A, z) := s A (z).
Note that dom ϕ = B × Bβ (0) thanks to the Lipschitz continuity of s A with Lipschitz
constant . Moreover, ϕ has the following properties:
(v) For each A ∈ B the mapping ϕ(A, ·) = s A is Lipschitz continuous on Bβ (0)
with Lipschitz constant ;
(vi) For each A ∈ B one has ϕ(A, 0) = x̄ = s A (0);
(vii) ϕ is continuous.
One needs to prove (vii) only. Let {An } be a sequence of operators in B which is
convergent to Ā and {z n } be a sequence of points from Bβ (0) convergent to z̄; then
Ā ∈ B and z̄ ∈ Bβ (0). Set ū = ϕ( Ā, z̄) and u n = ϕ(An , z n ) for each n ∈ N. For all n
we have
z n ∈ f (x̄) + An (u n − x̄) + F(u n ),
that is,
f (x̄) + Ā(u n − x̄) + F(u n ) z n + ( Ā − An )(u n − x̄).
Since each u n ∈ Bβ (x̄) and An → Ā as n → ∞, we obtain that
z n + ( Ā − An )(u n − x̄) ∈ B2β (0) for all n sufficiently large.
Then, using the definitions of u n and ū, property (v) reveals that
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A nonsmooth Robinson’s inverse function theorem
u n − ū
= s Ā (z n + ( Ā − An )(u n − x̄)) − s Ā (z̄)
≤ z n − z̄
+ 2 β
Ā − An → 0 as n → ∞.
Thus (vii) is established.
Step 4 Our next step is to show that
the mapping ( f + F)−1 has a nonempty-valued graphical localization around 0 for x̄.
(3)
In preparation for that, apply condition (a) to find δ such that for each two distinct
elements u and v of B3δ (x̄) there exists A ∈ B with
f (v) − f (u) − A(v − u)
< ε
v − u
.
(4)
Adjust δ if necessary to satisfy
0 < 6δ <
Clearly, δ < β. From (1),
β
.
(1/ + μ)
b := (1 − ε)δ < δ.
(5)
(6)
For any y ∈ Bεb (0), w ∈ Bδ (x̄), ũ ∈ Bδ (x̄) and A ∈ B the relations (5) and (6) yield
the estimate
y − f (w)+ f (x̄)+ A(w − ũ)
≤ y
+(ε+μ)
w − x̄
+μ
w − x̄
+μ
ũ − x̄
≤ εb + (ε + μ)δ + 2μδ < δ(2ε + 3μ)
< 3δ(1/ + μ) < β/2.
Hence,
y − f (w) + f (x̄) + A(w − ũ) ∈ Bβ/2 (0) whenever
(y, w, ũ, A) ∈ Bεb (0) × Bδ (x̄) × Bδ (x̄) × B.
(7)
Let y ∈ Bεb (0) be fixed. We will now find x ∈ ( f + F)−1 (y) ∩ Bδ (x̄); this will
prove (3). Denote
K = B2εδ (0)\{0}.
Fix u ∈ Bδ (x̄) and define the function
B A → u (A) = ϕ A, y − f (u) + f (x̄) + A(u − x̄) − u.
(8)
By (7) with ũ = x̄ and w = u, each value of the argument of ϕ in (8) is in dom ϕ, hence
dom u = B. Also note that, according to property (vii) above, for any u ∈ Bδ (x̄),
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R. Cibulka, A. L. Dontchev
the function u is continuous in its domain. If there exist A ∈ B and u ∈ Bδ (x̄) such
that u (A) = 0, then u ∈ ( f + F)−1 (y). From now on, assume that this is not the
case, that is,
(9)
u (A) = 0 for all A ∈ B and all u ∈ Bδ (x̄).
In further lines we will construct a sequence of points convergent to an x ∈ ( f +
F)−1 (y). To this end, we make use of the following two lemmas:
Lemma 2 Given u ∈ Bδ (x̄), suppose that there exist v ∈ Bδ (x̄)\{u} along with
à ∈ B satisfying
f (v) − f (u) − Ã(v − u)
≤ ε
v − u
and
f (v) + Ã(u − v) + F(u) y. (10)
Then u maps B into K . More precisely, we have
0 < u (A)
≤ ε
u − v
whenever A ∈ B.
(11)
Proof We show that (10) implies (11). Pick any A ∈ B. The first inequality follows
from (9). Then for z := y − f (v) + f (x̄) + A(u − x̄) − Ã(u − v), the inclusion (7)
with A := Ã, w := v and ũ = u combined with (5) implies that
z
≤ y − f (v)+ f (x̄)+ Ã(v − u)
+
A(u − x̄)
≤ β/2+μδ < β/2+β/6 < β.
Rearranging the inclusion in (10) gives us
f (x̄) + A(u − x̄) + F(u) y + f (x̄) + A(u − x̄) − f (v) − Ã(u − v) = z.
Therefore, u = ϕ(A, z) and we conclude that
u (A)
= ϕ A, y − f (u) + f (x̄) + A(u − x̄) − ϕ(A, z)
≤ f (v) − f (u) − Ã(v − u)
≤ ε
u − v
≤ 2εδ.
Hence (11) is satisfied. The last estimate also shows that u maps B into K .
Keeping u ∈ Bδ (x̄) fixed define the following set-valued mapping acting from K
into the subsets of B:
K h → u (h) = A ∈ B f (u + h) − f (u) − Ah
≤ ε
h
.
(12)
Lemma 3 Given u ∈ Bδ (x̄), suppose that u maps B into K . Then there exists
a continuous selection ψu of the mapping u such that the function defined as the
composition ψu ◦ u and acting from B into itself has a fixed point.
Proof Note that the values of u are closed convex sets. Fix any h ∈ K . Since ε < 1/,
we get that u + h − x̄
≤ u − x̄
+ h
≤ δ + 2εδ < 3δ. Hence u and u + h
are distinct elements of B3δ (x̄). Then (4) with v = u + h implies that u (h) = ∅.
Therefore dom u = K . We show next that u is inner semi-continuous. Fix any
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A nonsmooth Robinson’s inverse function theorem
h ∈ K , and let be an open set in B which has a non-empty intersection with u (h).
Pick any A from u (h) ∩ . According to (4), there is à ∈ B such that
f (u + h) − f (u) − Ãh
< ε
h
.
Since is open and A ∈ , there is λ ∈ (0, 1) such that Aλ := (1 − λ)A + λ Ã ∈ .
Put
V = τ ∈ K f (u + τ ) − f (u) − Aλ τ < ε
τ .
The estimate
f (u + h) − f (u) − Aλ h
≤ (1 − λ)
f (u + h) − f (u) − Ah
+ λ
f (u + h) − f (u) − Ãh
< (1 − λ)ε
h
+ λε
h
= ε
h
,
tells us that h ∈ V . Employing the continuity of f and the fact that Aλ is a continuous
linear mapping, we have that any τ sufficiently close to h belongs to V ; thus V is a
neighborhood of h in K . From the definitions of u and V we get that Aλ ∈ u (w)
for w ∈ V, therefore u (w) intersects for each w ∈ V . This proves that u is inner
semi-continuous. Michael’s selection theorem yields the existence of a continuous
selection ψu for u , that is, a continuous function acting from K into B with the
property that if w ∈ K and A := ψu (w) then
f (u + w) − f (u) − Aw
≤ ε
w
.
(13)
Since B is a compact convex set, u is a continuous function, and by assumption u
maps B into K , by the Brouwer’s fixed point theorem the composite mapping ψu ◦ u
acting from B into itself has a fixed point.
We will now employ an iterative procedure which generates sequences {xn } in X
and {An } in B whose entries have the following properties for each n ∈ N0 :
(i)
(ii)
(iii)
(iv)
xn − x̄
< δ;
0 < xn+1 − xn ≤ (lε)n x1 − x0 ;
f (xn+1 ) − f (xn ) − An (xn+1 − xn )
≤ ε
xn+1 − xn ;
f (xn ) + An (xn+1 − xn ) + F(xn+1 ) y.
Clearly, x0 := x̄ verifies (i) for n = 0. For any A ∈ B, we have x0 (A) = ϕ(A, y)−x0 .
Using the properties (vi) and (vii) along with (6) and (9), we get
0 < x0 (A)
= ϕ(A, y)−ϕ(A, 0)
≤ y
≤ εb < εδ < δ whenever A ∈ B.
Hence x0 maps B into K . According to Lemma 3, there exists a continuous selection
ψx0 of the mapping x0 such that the composite function ψx0 ◦ x0 has a fixed point.
Denote this fixed point by A0 ; that is, A0 = ψx0 (x0 (A0 )) ∈ B. Then
x1 := x0 + x0 (A0 ) = ϕ(A0 , y)
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R. Cibulka, A. L. Dontchev
satisfies (i) with n = 1, as well as (ii) and (iv) with n = 0. Since A0 = ψx0 (x1 − x0 ),
condition (iii) holds as well thanks to (13).
Further, we proceed by induction. Suppose that for a natural N > 0 we have found
xn+1 and An that satisfy conditions (i)–(iv) for all natural n < N . Set v := x N −1 and
à = A N −1 . Conditions (ii)–(iv) with n = N − 1 imply that the mapping x N satisfies
the assumption (10) of Lemma 2. Hence, by using Lemma 2 and then Lemma 3, there
exists A N ∈ B such that A N = ψx N (x N (A N )). Put
x N +1 = x N + x N (A N ) = ϕ A N , y − f (x N ) + f (x̄) + A N (x N − x̄) .
(14)
Then
f (x̄) + A N (x N +1 − x̄) + F(x N +1 ) y − f (x N ) + f (x̄) + A N (x N − x̄).
This is (iv) for n = N . Since A N = ψx N (x N +1 −x N ), (iii) holds for n = N . Combining
(11), (14), and (ii) for n = N − 1, gives us (ii) for n = N . Furthermore, since
x1 − x0 = x1 − x̄
= x0 (A0 )
≤ εb,
using (ii) and (6), we conclude that
x N +1 − x̄
≤
N
n=0
xn+1 − xn <
εb
x1 − x̄
≤
= εδ < δ.
1 − ε
1 − ε
We arrive at (i) for n = N + 1. The induction step is complete.
By (ii), the sequence {xn } is a Cauchy sequence, hence it converges to some x ∈
Bδ (x̄). Fix any n ∈ N. In view of (i), both xn and xn+1 are in Bδ (x̄) ⊂ Bβ (x̄).
Moreover, (7) for ũ := xn+1 , w := xn and A := An , along with (ii), implies that
y − f (xn ) + f (x̄) − An (xn+1 − xn )
< β/2 < β.
Using (iv) we obtain
xn+1 , y − f (xn ) − An (xn+1 − xn ) ∈ gph F ∩ Bβ (x̄) × Bβ (− f (x̄)) .
Passing to the limit and remembering the first of the key properties stated in the
beginning of Step 3, we conclude that f (x) + F(x) y, that is, x ∈ ( f + F)−1 (y) ∩
Bδ (x̄). Since y ∈ Bεb (0) was chosen arbitrarily,
Bεb (0) y → σ (y) := ( f + F)−1 (y) ∩ Bδ (x̄)
is a nonempty-valued localization of ( f + F)−1 ; that is, (3) is established.
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A nonsmooth Robinson’s inverse function theorem
Step 5 It remains to show that σ is a Lipschitz continuous function. Choose any
y , y ∈ Bεb (0). Pick any x ∈ σ (y ) and x ∈ σ (y ). Then there exists A ∈ B such
that
f (x ) − f (x ) − A(x − x )
≤ ε
x − x .
Then (7) reveals that both y − f (x ) + f (x̄) + A(x − x̄) and y − f (x ) + f (x̄) +
A(x − x̄) lie in Bβ (0). Moreover, we have
x = s A (y − f (x ) + f (x̄) + A(x − x̄))
and
x = s A (y − f (x ) + f (x̄) + A(x − x̄)).
Taking the difference, we obtain
x − x ≤ y − y + f (x ) − f (x ) − A(x − x )
≤ y − y + ε
x − x .
This gives us
x − x ≤
y − y .
1 − ε
Thus σ is both single-valued and Lipschitz continuous. The proof of Theorem 3 is
complete.
3 Final remarks
We will first comment on the proofs given by Izmailov [6] and Páles [10] and compare
them to our proof of Theorem 3.
The proof in [6] strongly relies on the fact that the spaces are finite dimensional.
The main step in that proof is an application of Brouwer’s fixed point theorem to a
function χ y on a ball Bδ (x̄) (using the notation in [6]) whose continuity is claimed
but never proved.1
Páles [10] uses in his proof Banach’s open mapping theorem, Michael’s selection
theorem, Ekeland’s variational principle and Kakutani’s fixed point theorem. We also
use Banach’s and Michael’s theorems, along with the general formulation in [3, Theorem 5F.1] of Robinson’s implicit function theorem but the whole idea of the proof is
different from [10]; the main step of the proof is an iterative procedure which resembles
Newton’s method.
We would like to also point out that our motivation to consider infinite dimensional
spaces is not based only on our desire to develop an alternative proof of Izmailov’s
1 Shortly before this paper was accepted for publication we received a letter by A. Izmailov where he
confirmed that indeed his proof is not complete and could be fixed by using some of the arguments in our
proof. He also noted that his proof heavily relies on the finite dimensions.
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R. Cibulka, A. L. Dontchev
theorem also covering the infinite-dimensional case. In a subsequent paper [11] Páles
gave an application of his theorem to optimal control; we do believe that our more
general result has the potential for further applications in that field. Furthermore,
there have been a number of developments in the last decade regarding Newton-type
methods applied to nonsmooth equations in infinite-dimensional spaces, e.g. in PDEconstrained optimization; some of them are broadly covered in the recent books by
Ito and Kunisch [5] and Ulbrich [14]. Strong regularity plays a prominent role in
these developments, as it has been since Josephy, a student of Robinson, applied it for
showing convergence of Newton’s method for variational inequalities. In particular,
Klatte and Kummer considered in [7, Section 10.1] nonsmooth equations where strong
regularity appears in the form of certain uniform invertibility of the elements of a set of
linear bounded operators which is closely related to but is different from the set A in our
paper. Note that Ulbrich [14] and Ito and Kunisch [5] consider variational inequalities
representing optimality conditions that are reduced to nonsmooth equations by using
e.g. the Fischer–Burmeister function. In our work we consider nonsmooth generalized
equations which occur typically in models of optimal control and in C 1,1 optimization.
We expect that applying our theorem and the analysis around it to specific nonsmooth
Newton-type methods in infinite dimensions will provide valuable contributions to the
area.
References
1. Clarke, F.H.: On the inverse function theorem. Pac. J. Math. 64, 97–102 (1976)
2. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
3. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings, 2nd edn. Springer,
Dordrecht (2014)
4. Ioffe, A.D.: Nonsmooth analysis: differential calculus of nondifferentiable mappings. Trans. Am. Math.
Soc. 266, 1–56 (1981)
5. Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. SIAM,
Philadelphia (2008)
6. Izmailov, A.F.: Strongly regular nonsmooth generalized equations. Math. Program. Ser. A 147, 581–
590 (2014)
7. Klatte, D., Kummer, B.: Nonsmooth equations in optimization. Regularity, calculus, methods and
applications. In: Nonconvex Optimization and Its Applications, vol. 60. Kluwer, Dordrecht (2002)
8. Kummer, B.: An implicit-function theorem for C 0,1 -equations and parametric C 1,1 -optimization. J.
Math. Anal. Appl. 158, 35–46 (1991)
9. Páles, Z.: Linear selections for set-valued functions and extensions of bilinear forms. Arch. Math. 62,
427–432 (1994)
10. Páles, Z.: Inverse and implicit function theorems for nonsmooth maps in Banach spaces. J. Math. Anal.
Appl. 209, 202–220 (1997)
11. Páles, Z.: On abstract control problems with non-smooth data. In: Recent Advances in Optimization,
pp. 205–216. Springer, Berlin (2006)
12. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)
13. Robinson, S.M.: An implicit-function theorem for a class of nonsmooth functions. Math. Oper. Res.
16, 292–309 (1991)
14. Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization
Problems in Function Spaces. SIAM, Philadelphia (2011)
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