Approximately convex functions and approximately monotone
operators
Huynh Van Ngai∗and Jean-Paul Penot†
Abstract
We present characterizations of some generalized convexity properties of functions with the help
of a general subdifferential. We stress the case of lower semicontinuous functions. We also study the
important case of marginal functions and we provide representation results.
Key words: approximate convexity, approximate monotonicity, paraconvexity, semiconvexity, subdifferential.
1
Introduction
Convexity or smoothness are often too strong assumptions for the needs of applications, for instance in
mathematical economics. Recently, much attention has been given to special classes of nonsmooth functions
having interesting properties which could serve as substitute to these assumptions. Among these classes
are the following ones: lower C 1 and lower C 2 functions ([59], [56], [1], [10, Thm 5.2], [43]), semismooth
functions ([34], [37]), subsmooth functions ([43], [16]), approximately convex functions in the sense of Green
[21], Hyers and Ulam [23], Páles [40], approximately convex functions in the sense of Luc-Ngai-Théra ([33],
[1], [14]), locally paraconvex (or weakly convex or semiconvex or hypoconvex, [61], [58], [8], [11]) functions.
Here we focus our attention on approximately convex functions. Most of the existing results characterizing
these classes have been established for locally Lipschitzian functions. It is our purpose here to tackle the
following questions:
1) Are such characterizations valid for lower semicontinuous (l.s.c.) functions?
2) Which subdifferentials can be used for characterizing such properties?
Question 1) is not without importance: in optimization it is useful to use l.s.c. functions, in particular
to substitute a constraint x ∈ F by the addition of the indicator function ιF to the objective function, the
indicator function ιF of a subset F of X being the function given by ιF (x) = 0 for x ∈ F, ιF (x) = +∞ for
x ∈ X\F ; obviously such a function is lower semicontinuous if F is closed, but it is not Lipschitzian (unless
F is empty or X). Although indicator functions are out of the scope of the present work, a related study of
approximately convex sets can be conducted (see [38]). Here we give a positive answer to question 1), what
requires a refined use of the mean value theorem. A partial answer to question 2) has been given in [10,
Thm 5.2] for the class of lower C 2 functions on a finite dimensional space and for the proximal, limiting
and Clarke subdifferentials. This result is extended in [36, Thm 3.6] to the class of approximately convex
functions and for some other subdifferentials (the Ioffe subdifferentials and the limiting subdifferential).
In [14], [15], [1], following the track of [59], [56], [27], [16], some monotonicity properties of subdifferentials
are used to give characterizations of some remarkable classes of functions. These properties are linked to
the question of integrating subdifferentials which is not considered here; we refer to [2], [16], [6], [27] and
∗ Department
† Laboratoire
of Mathematics, Pedagogical University of Quynhon, 170 An Duong Vuong, Qui Nhon, Vietnam
de Mathématiques Appliquées, CNRS UMR 5142, Faculté des Sciences, Av. de l’Université 64000 PAU,
France
1
their references). Here we endeavour to present a large class of subdifferentials for which a unified positive
answer can be provided.
A third question will be tackled elsewhere: which subdifferentials coincide on such classes of generalized
convex or generalized smooth functions?
2
Preliminaries
In the sequel, X is a Banach space with topological dual space X ∗ and F(X) denotes a subset of the set
S(X) of lower semicontinuous functions f : X → R∪{+∞}. The open (resp. clased) ball with center
x0 ∈ X and radius ρ > 0 is denoted by B(x0 , ρ) (resp. B(x0 , ρ)) and BX (resp. SX ) stands for B(0, 1)
(resp. B(0, 1)\B(0, 1)).
We will deal with the following classes of functions. The first one has been introduced by Vial [61] in
finite dimensions under the name of weak convexity; in Hilbert spaces, the terminologies semi-convex ([8],
[31]) and paraconvex ([58], [51]) are also used. Note that a function belongs to this class iff its index of
nonconvexity as defined in [50] is finite.
Definition 1 A function f : X → R∪{+∞} is said to be paraconvex (or semiconvex or weakly convex)
around x0 ∈ X if there exist γ, ρ > 0 such that for any x, y ∈ B(x0 , ρ) and any t ∈ [0, 1] one has
2
f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y) + γt(1 − t) kx − yk .
(1)
The following definition delineates a larger class; it has been introduced in [36] and used in [1], [14],
[15].
Definition 2 A function f : X → R∪{+∞} is said to be approximately convex around x0 ∈ X if for any
ε > 0 there exists δ > 0 such that for any x, y ∈ B(x0 , δ) and any t ∈ [0, 1] one has
f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y) + εt(1 − t) kx − yk .
(2)
In finite dimensions, these two classes have been characterized by a lower C 2 and a lower C 1 property,
respectively, in the sense of [56], [59] (see also [20], [43] for variants). Both classes have interesting stability
properties; see for instance [36, Prop. 3.1], [16, Section 6], [1] for the first assertion of the following
statement. The proof of the last one is easy.
Proposition 3 The set of functions f : X → R∪{+∞} which are approximately convex around x0 ∈ X
is a convex cone containing the functions which are strictly differentiable at x0 . It is stable under finite
suprema. Moreover, if f = h ◦ g, where g : X → Y is strictly differentiable at x0 and h : Y → R∪{+∞} is
approximately convex around g(x0 ), then f is approximately convex around x0 .
It can be shown that approximately convex functions retain some of the nice properties of convex
functions. In particular they are continuous on segments contained in their domains ([33, Cor. 3.3]) and
have radial derivatives ([33, Cor. 3.5]).
These two classes of functions can be related to some notions of generalized monotonicity via some
concepts of generalized derivatives or subdifferentials. We first recall these monotonicity properties. Here,
we adopt a new terminology, not just for the sake of a nice correspondence with the generalized convexity
notions recalled above, but essentially to avoid possible confusions: in [57], [15], [1], the original term
“strict” used in [59], [56], [16], [14] has been dropped in spite of the fact that a non strict notion was
already introduced in [59], [56]. This renunciation was motivated by the fact that there are useful strict
notions of convexity or generalized convexity which have nothing to do with the local property of strict
monotonicity; but it created some ambiguity. Other difficulties appear with the fact that in [20] the
term “submonotone” has been used in a distinct directional sense; this weaker notion will be given a
2
separate treatment and will not be considered here. Related notions of generalized cyclic monotonicity
are introduced or used in [27], [16], [15] for the needs of integration results but are out of the scope of the
present study.
Definition 4 A multivalued mapping T :X ⇒ X ∗ will be called approximately monotone (or submonotone)
around x0 ∈ dom(T ) provided that for each ε > 0 there exists ρ > 0 such that
hx∗1 − x∗2 , x1 − x2 i ≥ −εkx1 − x2 k
(3)
whenever xi ∈ B(x0 , ρ), x∗i ∈ T (xi ), i = 1, 2.
The following property is a strengthening of this notion which constitutes a classical example.
Definition 5 A multivalued mapping T :X ⇒ X ∗ will be called paramonotone (or hypomonotone) around
x0 ∈ dom(T ) provided there exist ρ, σ > 0 such that
hx∗1 − x∗2 , x1 − x2 i ≥ −σkx1 − x2 k2
(4)
whenever xi ∈ B(x0 , ρ), x∗i ∈ T (xi ), i = 1, 2.
We will relate the notion of approximate convexity to the notion of approximate monotonicity through
the use of subdifferentials. For the relationships between paraconvexity and paramonotonicity, we refer to
[39]. Here we adopt a very loose and versatile definition for the notion of subdifferential: a subdifferential
on a class F(X) of functions on a Banach space X will be just a correspondence ∂ : F(X) × X ⇒ X ∗
which assigns a subset ∂f (x) of the dual space X ∗ of X to any f ∈ F(X) and any x ∈ X at which f is
finite and which is such that 0 ∈ ∂f (x) if x is a local minimizer of f. This generality allows to encompass
a number of notions. Usually, some more conditions are imposed on ∂; we do not require them but, for
some implications, a mean value property described as follows is needed; it has a fuzziness character which
stems from the fact that it can be deduced from a form of trustworthiness (see [24], [32], [44], [62]).
Definition 6 (Fuzzy mean value theorem) A subdifferential ∂ is said to be valuable on X if for any x ∈ X,
y ∈ X\{x}, any lower semicontinuous function f : X → R∪{+∞} finite at x ∈ X and for any r ∈ R such
that f (y) ≥ r, there exist u ∈ [x, y) and sequences (un ) → u, (u∗n ) such that u∗n ∈ ∂f (un ), (f (un )) → f (u),
lim inf hu∗n , y − xi ≥ r − f (x),
(5)
n
r − f (x)
x − un
i≥
n
kx − uk
ky − xk
∗
lim kun k d(un , [x, y]) = 0.
lim inf hu∗n ,
∀x ∈ (x + R+ (y − x)) \[x, u),
n
(6)
(7)
When f is locally Lipschitz simpler forms of this result can be given (see [4, Thm 19] for the moderate
subdifferential, [30] for the Clarke subdifferential, [44] for the limiting subdifferential).
Many subdifferentials, but not all, can be introduced with the help of some notions of directional
derivative. Such tools can also be used for characterizations of generalized convexity properties. For that
reason, we recall some of them. In the sequel f is an element of F(X), x is a point in the domain of f and
v is a fixed vector of X.
The dag derivative of f is a rather special notion introduced in [42] (see also [45], [46], [22]) whose
interest seems to be limited to the fact that it is the largest possible notion which can be used in this
context. It is given by
f † (x, v) :=
1
(f (y + t(v + x − y)) − f (y)),
(t,y)→(0+ ,x),f (y)→f (x) t
lim sup
3
It majorizes both the upper radial (or upper Dini) derivative
1
0
f+
(x, v) := lim sup (f (x + tv) − f (x))
t→0+ t
and the Clarke-Rockafellar derivative or circa-derivative
f ↑ (x, v) := sup lim sup
inf
r>0 (t,y)→(0+ ,x) w∈B(v,r)
1
(f (y + tw) − f (y)).
t
f (y)→f (x)
When f is Lipschitzian, f ↑ coincides with the Clarke derivative f ◦ :
f ◦ (x, v) :=
1
(f (y + tw) − f (y)).
(t,y,w)→(0+ ,x,v) t
lim sup
The lower derivative (or contingent derivative or lower epiderivative or lower Hadamard derivative) is
given by
1
(f (x + tu) − f (x)).
f ! (x, v) := lim inf
(t,u)→(0+ ,v) t
It can also be denoted by f 0 (x, v) in view of its importance in optimization. Its radial variant fr! (x, v)
consisting in fixing u = v in the preceding lower limit is called the lower Dini derivative of f. The incident
derivative (or inner epiderivative, or adjacent derivative)
f i (x, v) := sup lim sup
r>0
t&0
inf
u∈B(v,r)
1
(f (x + tu) − f (x))
t
is intermediate between the contingent derivative and the circa-derivative. As the derivatives f ! , f ↑ it
is lower semicontinuous in v; as f ! it is bounded above by the upper derivative (or upper Hadamard
derivative, or upper hypo-derivative)
f ] (x, v) :=
1
(f (x + tw) − f (x)) = −(−f )! (x, v).
t
(t,w)→(0+ ,v)
lim sup
These derivatives can be compared and ordered. Several of them are such that their epigraphs are tangent
cones (in a related sense) to the epigraph of the function. Conversely, any notion of tangent cone T ? (see
[41] for a versatile construction) gives rise to a notion of generalized derivative f ? given by
f ? (x, v) := inf{r ∈ R : (v, r) ∈ T ? (Ef , xf )},
where Ef is the epigraph of f and xf := (x, f (x)). In turn, to any directional derivative f ? one can
associate a subdifferential via the relation
∂ ? f (x) := {x∗ ∈ X ∗ : x∗ ≤ f ? (x, ·)}.
Note that if N ? (Ef , xf ) is the normal cone to the epigraph Ef of f at xf , i.e. the polar cone of T ? (Ef , xf ),
one has
x∗ ∈ ∂ ? f (x) ⇔ (x∗ , −1) ∈ N ? (Ef , xf ).
However, not all interesting notions of subdifferential arise from a directional derivative or a notion of
tangent cone. Some of them, such as the Fréchet subdifferential ∂ − (or ∂ F ) or the limiting subdifferential
∂ (or ∂ L ) are directly associated to a notion of normal cone or are defined in an analytic way (see [26] for
instance).
4
3
Characterizations of approximate convexity
We first present characterizations of approximate convexity around a point; they generalize previous results
in [14, Thm 2], [1, Thm 4.5] to the non Lipschitzian case and to a general subdifferential.
Theorem 7 Let x0 ∈ X, and let f be lower semicontinuous. Suppose the subdifferential ∂f of f is
contained in ∂ † f. Then, among the following assertions, one has the implications (a)⇒(b)⇒(c)⇔(c’)⇒(d).
If moreover ∂ is valuable on X, all these assertions are equivalent.
(a) f is approximately convex around x0 ;
(b) for any ε > 0 there exists ρ > 0 such that for any x ∈ B(x0 , ρ) and any v ∈ B(0, ρ) one has
f † (x, v) ≤ f (x + v) − f (x) + ε kvk ;
(8)
(c) for any ε > 0 there exists ρ > 0 such that for any x ∈ B(x0 , ρ), any x∗ ∈ ∂f (x) and any
(u, t) ∈ SX × (0, ρ) one has
f (x + tu) − f (x)
+ ε;
(9)
hx∗ , ui ≤
t
(c’) for any ε > 0 there exists ρ > 0 such that for any x ∈ B(x0 , ρ), any x∗ ∈ ∂f (x) and any v ∈ B(0, ρ)
one has
hx∗ , vi ≤ f (x + v) − f (x) + ε kvk ;
(d) ∂f is approximately monotone around x0 .
Proof. (a)⇒(b) Given ε > 0, let δ > 0 be such that for any y, z ∈ B(x0 , δ), t ∈ (0, 1) one has
f ((1 − t)y + tz) ≤ (1 − t)f (y) + tf (z) + εt(1 − t) ky − zk .
Given x ∈ B(x0 , δ), let v ∈ X be such that z := x + v ∈ B(x0 , δ); this is ensured by taking x ∈ B(x0 , ρ),
v ∈ B(0, ρ) with ρ = δ/2. Then for y ∈ X close enough to x we have y ∈ B(x0 , δ) and
f ((1 − t)y + t(x + v)) − f (y)
≤ f (x + v) − f (y) + ε(1 − t) ky − (x + v)k ,
t
so that, passing to the limit superior as (t, y, f (y)) → (0+ , x, f (x)), since lim supy→x −f (y) ≤ −f (x), we
get relation (8).
(b)⇒(c) Given any x∗ ∈ ∂f (x) ⊂ ∂ † f (x) and any (u, t) ∈ SX × (0, ρ), setting v = tu we get (9) from
the relation x∗ ≤ f † (x, ·).
(c)⇔(c’) is obtained by changing v into tu with t = kvk , u ∈ SX and vice versa.
(c)⇒(d) is proved in [14, Thm 2] and [1, Thm 4.5] for the Clarke subdifferential; the general case is
similar: given ε > 0, taking ρ > 0 as in (c), for any x, y ∈ B(x0 , ρ/2), any x∗ ∈ ∂f (x), y ∗ ∈ ∂f (y), setting
t := kx − yk , u := t−1 (y − x), we get from (9) the first inequality below, while the second one is obtained
by changing (x, y, x∗ ) into (y, x, y ∗ )
hx∗ , y − xi ≤ f (y) − f (x) + ε kx − yk ,
hy ∗ , x − yi ≤ f (x) − f (y) + ε kx − yk ,
hence hx∗ − y ∗ , x − yi ≥ −2ε kx − yk .
(d)⇒(a) when ∂ is valuable. Assume ∂f is approximately monotone around x0 : for any ε > 0 one can
find ρ > 0 such that for any x, y ∈ B(x0 , ρ), any x∗ ∈ ∂f (x), y ∗ ∈ ∂f (y) one has
hx∗ − y ∗ , x − yi ≥ −ε kx − yk .
5
(10)
Let δ := ρ and let x, y ∈ B(x0 , δ), t ∈ (0, 1), z := tx + (1 − t)y. Without loss of generality, to prove (2) we
may assume that x 6= y and f (x), f (y) < +∞. Applying Definition 6 to f on [y, z] with r < f (z), we get
u ∈ [y, z) and sequences (un ) → u, (u∗n ) such that u∗n ∈ ∂f (un ) for each n and
lim inf hu∗n ,
n
r − f (y)
x − un
i>
.
kx − un k
kz − yk
(11)
Let s ∈ (0, 1) be such that z = x+s(u−x) and let zn = x+s(un −x). Since (un ) → u, one has (zn ) → z and
for n larger than a certain k ∈ N one has r < f (zn ) as f is l.s.c. at z. Moreover kzn − xk = (1 − tn ) ky − xk
for some sequence (tn ) → t. Applying again Definition 6 to f on [x, zn ], we obtain vn ∈ [x, zn ) and
∗
∗
sequences (vn,p ) → vn , (vn,p
) such that vn,p
∈ ∂f (vn,p ) for any n, p and
∗
lim inf hvn,p
,
p
r − f (x)
r − f (x)
un − vn,p
i>
=
.
kun − vn,p k
kzn − xk
(1 − tn ) ky − xk
(12)
From relation (11) there exists some m ≥ k such that for n ≥ m one has [x, un ] ⊂ B(x0 , δ),
hu∗n ,
x − un
r − f (y)
r − f (y)
vn − un
i = hu∗n ,
i>
=
.
kvn − un k
kx − un k
ky − zk
t ky − xk
(13)
On the other hand, by (13) and (12), one can find some q(n) such that for p ≥ q(n) one has
vn,p − un
r − f (y)
i>
,
kvn,p − un k
t ky − xk
r − f (x)
un − vn,p
∗
i>
.
hvn,p
,
kun − vn,p k
(1 − tn ) ky − xk
hu∗n ,
since for each n, (vn,p ) → vn . Adding the corresponding sides of these inequalities and using relation (10),
one gets
r − f (y)
r − f (x)
ε≥
+
.
t ky − xk (1 − tn ) ky − xk
Passing to the limit as n → ∞, one deduces
εt(1 − t) ky − xk ≥ (1 − t)(r − f (y)) + t (r − f (x)) .
Since r is arbitrarily close to f (z), one obtains (2).
Remark. The proof we gave shows that a quantitative form of the result can be given. Reformulating
assertion (b) as
(b’) for any ε > 0 there exists ρ > 0 such that for any x ∈ B(x0 , ρ), and any v ∈ X such that
x + v ∈ B(x0 , ρ) relation (8) holds,
we see that we can take ρ = δ, where δ > 0 is as in (2). This fact incites to introduce the gage of
approximate convexity of f at x0 as the function α : (0, +∞) → (0, +∞) given by
α(ε) := sup{ρ : (8) holds}.
Its quasi-inverse αe given by
αe (ρ) := inf{
tf (x) + (1 − t)f (y) − f (tx + (1 − t)y)
: t ∈ (0, 1), x, y ∈ B(x0 , ρ)}
t(1 − t) kx − yk
can be called the modulus of approximate convexity of f at x0 . It can be computed with the help of relation
(8).
Corollary 8 The assertions (a), (b), (c), (d) of the preceding theorem are equivalent whenever
(i) X is an arbitrary Banach space and ∂ is the Clarke or the Ioffe subdifferential;
(ii) X is an Asplund space and ∂ is the Fréchet subdifferential or the Hadamard subdifferential.
Moreover, they are equivalent to the variant of assertion (b) obtained by replacing f † by f ↑ (and, if X
is an Asplund space, by f ! ).
6
4
Marginal functions
In this section we present conditions ensuring that a marginal function is approximately convex. Such
functions are important in optimization. In the following section we will look for a converse. The marginal
function we consider is of the form
x ∈ X0 ,
f (x) := sup g(s, x)
s∈S
where S is an arbitrary set, X0 is an open subset of a Banach space X and g : S × X0 → R. In the sequel,
for x ∈ X0 , ε > 0, we define the set of ε-approximate minimizers of gs := g(·, x) by
M (x, ε) := {s ∈ S : g(s, x) ≥ f (x) − ε}
{s ∈ S : g(s, x) ≥ ε−1 }
if f (x) < +∞
if f (x) = +∞.
Given a valuable subdifferential ∂ on F(X), and s ∈ S, x ∈ X0 , v ∈ X, we set ∂g(s, x) := ∂gs (x) and
hs (x, v) := h(s, x, v) := sup{hx∗ , vi : x∗ ∈ ∂g(s, x)}
If ∂g(s, x) = ∅, we set hs (x, v) = −∞.
The assumptions of the following result have a technical character. However, they are reminiscent to
conditions used in other studies. They will be transformed in more practical hypothesis in the last part of
this section.
Theorem 9 Let X0 be an open subset of a Banach space X and let x0 ∈ X0 . Under the following
assumptions, the function f is approximately convex around x0 .
(H1 ) For every s ∈ S, gs := g(s, .) is lower semicontinuous.
(H2 ) There is a neighborhood V of x0 in X0 such that for all x ∈ V , (xn ) → x, (αn ) → 0+ , there exist
sn ∈ M (xn , αn ) with (g(sn , x)) → f (x).
(H3 ) There is a neighborhood W of x0 such that for all (x, u) ∈ W ×BX , for every (xn ) → x, (αn ) → 0+ ,
sn ∈ S with sn ∈ M (x, αn ) for each n ∈ N then either lim inf n h(sn , xn , u) = −∞ or one has
lim inf [h(sn , xn , u) − h(sn , x, u)] 6 0.
n→∞
(H4 ) For any ε > 0, one can find δ > 0 such that for all (x, u, t) ∈ B(x0 , δ) × BX × (0, δ) there exists
γ > 0 such that for every s ∈ M (x, γ) one has
hs (x, u) 6
1
(g(s, x + tu) − g(s, x)) + ε.
t
Proof. In view of Corollary 8 it suffices to show that for any ε > 0 there exists δ > 0 such that
f ↑ (x, u) 6
∀(x, u, t) ∈ B(x0 , δ) × BX × (0, δ)
1
(f (x + tu) − f (x)) + ε.
t
Given ε > 0, let us take δ > 0 as in (H4 ) and such that B(x0 , δ) ⊂ V ∩ W . Let (x, u, t) ∈ B(x0 , δ) × BX ×
(0, δ). Let γ be associated to (x, u, t) as in (H4 ). By definition of f ↑ (x, u), using the fact that f is lower
semicontinuous by (H1 ), it suffices to show that for any ρ > 0 one has
lim sup
inf
1
(y,f (y),t)→(x,f (x),0+ ) v∈B(u,ρ) t
(f (y + tv) − f (y)) 6
1
(f (x + tu) − f (x)) + ε.
t
Let (xn ) → x, (tn ) → 0+ be such that (f (xn )) → f (x) and
lim
inf
1
n→∞ v∈B(u,ρ) tn
(f (xn + tn v) − f (xn )) =
1
(f (y + tv) − f (y)) .
(y,f (y),t)→(x,f (x),0+ ) v∈B(u,ρ) t
lim sup
7
inf
(14)
By (H2 ) we can pick sn ∈ M (xn + tn u, t2n ) such that (g(sn , x)) → f (x). By the mean value theorem
applied to the function g(sn , .) on [xn , xn + tn u], we can find zn ∈ B([xn , xn + tn u], tn ), zn∗ ∈ ∂g(sn , zn )
such that
1
(g(sn , xn + tn u) − g(sn , xn )) 6 hzn∗ , ui + tn 6 hsn (zn , u) + tn .
(15)
tn
Since (xn ) → x, (tn ) → 0+ , we have (zn ) → x. Let us first suppose that (hsn (zn , u)) is not bounded below.
Then, for each n ∈ N, we have
1
1
(f (xn + tn v) − f (xn )) 6
(f (xn + tn u) − f (xn ))
tn
v∈B(u,ρ) tn
1
(g(sn , xn + tn u) − g(sn , xn )) + tn
6
tn
inf
and, by our choice of (xn ), (tn ) , inequality (14) holds, the left hand side being −∞. Now, when (hsn (zn , u))
is bounded below, by assumption (H3 ) we have
lim inf [hsn (zn , u) − hsn (x, u)] 6 0
n→∞
and we can find an infinite subset K of N and a sequence (γn ) → 0 such that
hsn (zn , u) 6 hsn (x, u) + γn
∀n ∈ K.
(16)
Since (g(sn , x)) → f (x), without loss of generality we may assume that sn ∈ M (x, γ) for every n ∈ K.
Therefore, by (H4 ), we obtain
1
(g(sn , x + tu) − g(sn , x)) + ε
t
1
6 (f (x + tu) − g(sn , x)) + ε
t
hsn (x, u) 6
Hence, for all n ∈ K, in view of the choice of sn in M (xn + tn u, t2n ) and of relations (15), (16) we get
1
1
(f (xn + tn v) − f (xn )) 6
(f (xn + tn u) − f (xn ))
tn
v∈B(u,ρ) tn
1
6
(g(sn , xn + tn u) − g(sn , xn )) + tn
tn
6 hsn (zn , u) + 2tn 6 hsn (x, u) + γn + 2tn
1
6 (f (x + tu) − g(sn , x)) + ε + γn + 2tn .
t
inf
Passing to limit as n → ∞, we obtain inequality (14).
Let us give some simple criteria ensuring conditions (H2 )-(H4 ).
Lemma 10 Condition (H2 ) is satisfied whenever f is l.s.c. and one of the following conditions holds:
(H20 ) There is a neighborhood V of x0 in X0 such that for any x ∈ V , α > 0 there exists β > 0 such
that M (y, β) ⊂ M (x, α) for any y ∈ B(x, β).
(A2 ) There is a neighborhood V of x0 in X0 such that for any x ∈ V , α > 0 there exists β > 0 such
that gs (y) ≤ gs (x) + α for any y ∈ B(x, β) and all s ∈ M (y, β).
In particular, (H2 ) is satisfied whenever f is l.s.c. on some neighborhood V of x0 and gs is upper
semicontinuous on V, uniformly with respect to s ∈ S.
8
Proof. The first assertion is obvious: given x ∈ V , (xn ) → x, (αn ) → 0, for any sn ∈ M (xn , αn ),
ε > 0, taking β > 0 associated with α := ε as in (H20 ), one can find m ∈ N such that xn ∈ B(x, β), αn ≤ β
for n ≥ m, so that sn ∈ M (xn , αn ) ⊂ M (x, ε) and g(sn , x) ≥ f (x) − ε for n ≥ m; since g(sn , x) ≤ f (x) for
each n, and since ε is an arbitrary positive number, we get (g(sn , x)) → f (x).
Now, let us prove that (A2 ) implies (H20 ). Given x ∈ V , α > 0, let β ∈ (0, α/2) be such that
B(x, β) ⊂ V , f (y) > f (x) − α/4 and g(s, y) < g(s, x) + α/4 for every y ∈ B(x, β) and s ∈ M (y, β). Then,
for y ∈ B(x, β) and s ∈ M (y, β), we have
g(s, x) > g(s, y) − α/4 ≥ f (y) − β − α/4 ≥ f (x) − α/4 − β − α/4 ≥ f (x) − α,
so that s ∈ M (x, α).
Finally, the assumptions of the last assertion clearly imply (A2 ).
The following criterion is a generalization of [19], [20, Lemma 5.3]; its proof follows ideas of [19], [20]
and [48]. Here we set
M (x, y, ε) :=
{s ∈ S : g(s, x) ≥ f (y) − ε} if f (y) < +∞
{s ∈ S : g(s, x) ≥ ε−1 } if f (y) = +∞.
Proposition 11 Suppose that for each s ∈ S the function gs := g(s, ·) is continuous. Given a valuable
subdifferential ∂ and x0 ∈ dom f , suppose that the following condition is satisfied:
(B2 ) there exist δ > 0 and m > 0 such that
{x∗ : x∗ ∈ ∂gs (x), x ∈ B(x0 , δ), s ∈ M (x, x0 , δ)} ⊂ mBX ∗ .
Then f is Lipschitzian with rate m around x0 and conditions (A2 ), (H2 ) are satisfied.
Proof. Let γ ∈ (0, (m + 2)−1 δ) with γ −1 > f (x0 ). Since f is lower semicontinuous, there is some
ρ ∈ (0, γ/2) such that f (y) > f (x0 ) − γ for each y ∈ V := B(x0 , ρ). For any x, y ∈ B(x0 , ρ) and any
s ∈ M (y, γ) one has
g(s, y) > f (x0 ) − 2γ > f (x0 ) − δ.
Setting h(t) := g(s, y + t(x − y)),
t := max{t ∈ [0, 1] : ∀r ∈ [0, t] h(r) ≥ g(s, y) − mγ},
the mean value property yields z ∈ [y + t(x − y), y), a sequence (zn ) → z and zn∗ ∈ ∂gs (zn ) such that
lim inf hzn∗ , t(y − x)i ≥ h(0) − h(t).
n
(17)
Since g(s, z) ≥ g(s, y) − mγ > f (x0 ) − 2γ − mγ ≥ f (x0 ) − δ, we have g(s, zn ) > f (x0 ) − δ for n large
enough. Our assumption ensures that kzn∗ k ≤ m for each such n. Then, if t < 1, by continuity, we have
h(t) = g(s, y) − mγ = h(0) − mγ,
mγ > 2mρ ≥ mt ky − xk ≥ lim inf hzn∗ , t(y − x)i ≥ h(0) − h(t) = mγ,
n
a contradiction. Thus t = 1 and g(s, u) ≥ g(s, y) − mγ > f (x0 ) − δ for each u ∈ [x, y]. Thus relation (17)
and the inequalities kzn∗ k ≤ m yield
m ky − xk ≥ h(0) − h(1) = g(s, y) − g(s, x).
(18)
Taking the supremum over s ∈ M (y, γ), we get f (y) − f (x) ≤ m ky − xk . Choosing x := x0 we see that
f (y) < +∞. Since x, y are arbitrary in B(x0 , ρ) we obtain that f is Lipschitzian with rate m on B(x0 , ρ).
9
Now let us show that (A2 ) is satisfied (hence (H2 ) is satisfied too, by the preceding lemma). Given
α > 0 and x ∈ B(x0 , ρ), let β ∈ (0, γ) be such that β < ρ − kx − x0 k and β < α/m. Then, for y ∈ B(x, β),
s ∈ M (y, β) ⊂ M (y, γ), relation (18) ensures that
g(s, x) ≥ g(s, y) − m ky − xk ≥ g(s, y) − mβ ≥ g(s, y) − α.
Now let us give a criterion ensuring (H3 ).
Lemma 12 Suppose S is a topological space, the functions gs are differentiable on some neighborhood
W of x0 and that for each u ∈ X the mapping (s, x) 7→ gs0 (x)(u) is continuous on S × W. Suppose
moreover that the maximization of g(·, x) is well-posed in the following sense: for all (xn ) → x, (αn ) → 0,
sn ∈ M (xn , αn ), the sequence (sn ) has a converging subsequence. Then condition (H3 ) is satisfied.
Note that the well-posedness assumption is satisfied whenever S is compact.
Proof. Let x ∈ V , (xn ) → x, (αn ) → 0, sn ∈ M (xn , αn ). By assumption, taking a subsequence
if necessary, we may assume
(sn ) converges to some s ∈ S. By our continuity assumption, we have then
(hsn (xn , u)) = gs0 n (xn )(u) → gs0 (x)(u) and (hsn (x, u)) = gs0 n (x)(u) → gs0 (x)(u), so that (H3 ) is satisfied.
Finally, let us give a criterion ensuring (H4 ).
Lemma 13 Suppose the functions gs are differentiable on some neighborhood V of x0 and that the family
(gs0 )s∈S is equicontinuous at x0 . Then condition (H4 ) is satisfied.
Proof. We may suppose V is some ball B(x0 , ρ) with ρ > 0. Given ε > 0, let δ ∈ (0, ρ/2) be such that
kgs0 (x) − gs0 (x0 )k < ε/2 for (s, x) ∈ S × B(x0 , 2δ). Then for (x, s, u, t) ∈ B(x0 , δ) × S × BX × (0, δ) we can
find some θ ∈ (0, 1) such that
gs0 (x)(u) −
1
(gs (x + tu) − gs (x)) = gs0 (x)(u) − gs0 (x + θtu)(u)
t
≤ kgs0 (x) − gs0 (x0 )k + kgs0 (x + θtu) − gs0 (x0 )k < ε.
The following theorem gives refined conditions which ensures conditions (H2 ) and (H3 ). Part of it is
akin to [48, Cor. 3.11]. These conditions involve the subdifferential ∂g of g or the derivative of g with
respect to x when the latter is assumed to exist.
Theorem 14 Suppose that for every s ∈ S, gs is continuous, that (H1 ) and (B2 ) hold and that the
following condition is satisfied:
(A3 ) For each x in some neighborhood W of x0 , for each ε > 0, there exists δ > 0 such that
∂g(s, y) ⊆ ∂g(s, x) + εBX ∗
for all y ∈ B(x, δ) and s ∈ M (y, x, δ) ∩ M (x, δ).
Then (H2 ) and (H3 ) are satisfied.
As a result, if, in addition, (H4 ) is satisfied, then f is approximately convex at x0 .
Proof. Relation (18) in the proof of Proposition 11 shows that there exist m > 0, ρ > 0, γ > 0 such
that for all x ∈ B(x0 , ρ), y ∈ B(x0 , ρ) one has
g(s, x) 6 g(s, y) + m||x − y|| for all s ∈ M (x, γ).
10
Let us show that (H3 ) holds. Given ε > 0 let us choose δ > 0 as in (A3 ), with δ ≤ ρ. Diminishing
γ if necessary, we may suppose γ < δ/(m + 1) and γ < ρ − kx − x0 k . Then, for any x ∈ B(x0 , ρ),
y ∈ B(x, γ), s ∈ M (x, γ), we have y ∈ B(x0 , ρ) and
f (x) − δ ≤ f (x) − γ(m + 1) ≤ g(s, x) − m||x − y|| 6 g(s, y).
Thus s ∈ M (x, δ) ∩ M (y, x, δ) that is M (x, γ) ⊆ M (x, δ) ∩ M (y, x, δ). Therefore, from (A3 ), we obtain
that for any ε > 0, x ∈ B(x0 , ρ), there exists γ > 0 such that
∂g(s, y) ⊂ ∂g(s, x) + εBX ∗
for all y ∈ B(x, γ), s ∈ M (x, γ). This implies (H3 ): for every u ∈ BX , (xn ) → x, (αn ) → 0+ , sn ∈ S with
sn ∈ M (x, αn ) for each n ∈ N one has h(sn , xn , u) 6 h(sn , x, u) + ε for n large enough.
Corollary 15 If for each s ∈ S the function gs is Fréchet differentiable on some neighborhood V of x0 , if
the family (gs0 ) is equicontinuous at each point of V and if the set {gs0 (x0 ) : s ∈ S} is bounded in X ∗ then
f is approximately convex at x0 .
Proof. Assumption (H4 ) is satisfied by Lemma 13. The equicontinuity property ensures assumption
(A3 ) since for each x ∈ V and any ε > 0 we can find δ := δ(x, ε) > 0 such that B(x, δ) ⊂ V and
kgs0 (y) − gs0 (x)k < ε for s ∈ S, y ∈ B(x, δ). Now, taking ε = 1 and setting r := sup{kgs0 (x0 )k : s ∈ S}, for
s ∈ S, y ∈ B(x0 , δ(x0 , 1)) we have kgs0 (y)k ≤ r + 1, so that (B2 ) is satisfied.
5
Representation of approximately convex functions
Now let us turn to a representation of approximately convex functions by marginal functions. We need
some preliminary results.
Recall that a function f : X → R is said to be uniformly (Fréchet) differentiable on a subset A of X if
f is Fréchet differentiable at each x ∈ A and for any ε > 0, there exists δ > 0 such that
∀(x, v) ∈ A × δBX
|f (x + v) − f (x) − f 0 (x)(v)| ≤ ε kvk .
(19)
Lemma 16 Let f : X → R be Lipschitzian and uniformly differentiable on the ball B(0, ρ) for some ρ > 0.
Given σ, τ ∈ (0, ρ) with σ + τ < ρ, let U := B(0, τ ), fu (x) := f (x − u) for u ∈ U, x ∈ B(0, σ). Then the
family (fu0 )u∈U is equicontinuous on B(0, σ).
Proof. Given x ∈ B(0, σ), ε > 0, we want to find some η > 0 such that
kfu0 (x) − fu0 (y)k < 4ε
∀y ∈ B(x, η), u ∈ U.
(20)
Let ` be the Lipschitz rate of f on B(0, ρ), let δ > 0 be related to ε as in (19) and let ξ = min(δ, ρ − σ − τ ).
Given u ∈ U , v ∈ ξBX , y ∈ B(x, η) with η > 0, η ≤ σ − kxk , η ≤ εξ/`, we use the inequalities
|f 0 (x − u)v − f 0 (y − u)v| ≤ |f (x − u + v) − f (x − u) − f 0 (x − u)(v)| + |f (x − u + v) − f (y − u + v)|
+ |f (x − u) − f (y − u)| + |f (y − u + v) − f (y − u) − f 0 (y − u)(v)|
≤ 2ε kvk + 2` kx − yk ≤ 2εξ + 2`η ≤ 4εξ.
They show that (20) holds.
11
Lemma 17 Let ϕ ∈ C 1 (R, R) with ϕ0 (0) = 0 and let h : X → R be continuous on B(0, ρ), continuously
differentiable on B(0, ρ)\{0}, and such that h0 is bounded around 0, with h(0) = 0. Then ϕ ◦ h : X → R is
0
continuously differentiable on B(0, ρ). If h0 is bounded on B(0, ρ)\{0} then (ϕ ◦ h) is bounded on B(0, ρ).
If h satisfies the following assumptions, then, for each c ∈ (0, 1), the function ϕ◦h : X → R is uniformly
differentiable on B(0, cρ):
(P1 ) h is continuous on B(0, ρ) and, for any γ ∈ (0, ρ), h is uniformly differentiable on B(0, ρ)\B(0, γ).
(P2 ) h0 (.) is bounded on B(0, ρ) \ {0}.
Proof. Let ` > 0 and σ ∈ (0, ρ) be such that |h0 (x)| ≤ ` for x ∈ B(0, σ)\{0}. Then for any w, x ∈
B(0, σ)\{0}, using the mean value property, we see that |h(x) − h(w)| ≤ ` kx − wk (this is obvious if
0∈
/ [w, x], and the inequality is obtained by continuity if 0 ∈ [w, x]). Taking the limit as w → 0, we get
|h(x)| ≤ ` kxk for x ∈ B(0, σ). Given ε > 0, we can find some α > 0 such that |ϕ(t)| ≤ |t| ε/` for |t| ≤ α.
Then, for x ∈ B(0, η), with η := min(σ, α/`) we have |ϕ(h(x))| ≤ ε kxk . Thus ϕ ◦ h is differentiable at 0,
with (ϕ◦h)0 (0) = 0. Since (ϕ◦h)0 (x) = ϕ0 (h(x))h0 (x) for x ∈ B(0, ρ)\{0} and h0 is bounded on B(0, σ)\{0}
0
and ϕ0 (t) → 0 as t → 0, we get that (ϕ ◦ h) is continuous at 0. The second assertion is proved similarly.
Let us prove the last assertion. By assumption (P2 ), there is some m > 0 such that ||h0 (x)|| 6 m for each
x ∈ B(0, ρ)\{0} and, as above, we get that ||h(w)−h(x)|| 6 m kw − xk for any w, x ∈ B(0, ρ). In particular,
h(B(0, ρ)) is bounded. Let a > 0 be such that h(B(0, ρ)) ⊆ [−a, a]. Let b := sup{|ϕ0 (t)| : t ∈ [−a, a]};
without loss of generality we may suppose m ≥ b. Since ϕ ∈ C 1 (R, R), for any ε > 0, there exists δε > 0
such that
ε
|s|
(21)
ϕ(r + s) − ϕ(r) − ϕ0 (r)(s) ≤
3m
for all r ∈ [−a, a], s ∈ (−δε , δε ).
Since ϕ0 (r) → 0 as r → 0, and since h is continuous at 0, there exists γ > 0 such that
ε
0
, ∀x ∈ B(0, γ).
(22)
ϕ (h(x)) <
3m
By assumption (P1 ), h is uniformly Fréchet differentiable on A := B(0, ρ) \ B(0, γ), so that, given ε > 0,
there exists δε0 > 0 such that
ε
∀x ∈ A, v ∈ δε0 BX
kvk ,
(23)
h(x + v) − h(x) − h0 (x)(v) <
3m
Let us set δ = min{ρ − cρ, δε /m, δε0 }. Given x ∈ B(0, cρ)\{0}, v ∈ δBX , we have the following estimate:
(24)
(ϕ ◦ h)(x + v) − (ϕ ◦ h)(x) − (ϕ ◦ h)0 (x).v 6
6 |ϕ(h(x + v)) − ϕ(h(x)) − ϕ0 (h(x))(h(x + v) − h(x))| + |ϕ0 (h(x))|h(x + v) − h(x) − h0 (x)(v)
Since |h(x + v) − h(x)| 6 m kvk < δε , as x + v and x belong to B(0, ρ), by (21) we obtain
|ϕ(h(x + v)) − ϕ(h(x)) − ϕ0 (h(x))(h(x + v) − h(x))| 6
ε
ε
|h(x + v) − h(x)| 6 kvk ;
3m
3
(25)
ε
ε
kvk ≤ kvk
3m
3
(26)
on the other hand, when ||x|| ≥ γ then, by (23)
|ϕ0 (h(x))||h(x + v) − h(x) − h0 (x)(v)| 6 b
and when ||x|| < γ, by (22)
|ϕ0 (h(x))|. (|h(x + v) − h(x) |+| h0 (x)(v)|) 6
12
ε
2ε
(m kvk + m kvk) ≤
kvk .
3m
3
(27)
From (24), (25), (26) and (27), we get
|(ϕ ◦ h)(x + v) − (ϕ ◦ h)(x) − (ϕ ◦ h)0 (x)(v)| 6 ε kvk
for all x ∈ B(0, cρ)\{0}, v ∈ δBX and, since ϕ ◦ h is differentiable at 0, the proof is complete.
Remark: If the norm of X is uniformly Fréchet differentiable on the unit sphere then h(·) = || · ||
satisfies properties (P1 )-(P2 ) of the lemma.
The following proposition requires simple adaptations of [47, Prop. 2] (which is similar to [18] Prop.
5.1 p. 57 and Lemma 1.3 p. 340 although the conclusions are different). It enables one to use a smoothness
assumption on the space which is weaker than requiring that the norm is Fréchet differentiable off {0} : it
assumes that the space has a Fréchet differentiable bump function. Here we say that a function b on X is
a bump function if it takes its values in [0, 1], if b(0) = 1 and if b is null outside the unit ball BX .
Proposition 18 Let X be a normed space on which there exists a locally Lipschitz, continuously differentiable bump function. Then, for each c > 1, there exists a function h : X → R which is continuously
differentiable on X\{0} and such that
kxk ≤ h(x) ≤ ckxk
∀x ∈ X.
Such a function will be called a potential.
We also need a refinement of a well-known density property.
Lemma 19 Let ∂ be a valuable subdifferential on X and let f : X → R∪{+∞} be an arbitrary lower
semicontinuous function. Then for any x ∈ dom f there exists a sequence (xn , x∗n ) in the graph of ∂f such
that (xn ) → x, and lim inf n hx∗n , x − xn i ≥ 0.
Proof. When x is a local minimizer of f the result is obvious (take xn = x, x∗n = 0 for each n ∈ N).
Suppose x is not a local minimizer of f ; then there exists a sequence (un ) → x such that f (un ) < f (x)
for each n ∈ N (hence un 6= x for each n). Applying the mean value theorem to f on [un , x] we get some
(xn , x∗n ) ∈ ∂f such that kx − xn k < 2 kx − un k (since xn can be arbitrarily close to some point of [un , x))
and
kx − xn k
1
hx∗n , x − xn i ≥
(f (x) − f (un )) − .
kx − un k
n
Since (f (x) − f (un )) ≥ 0, we get lim inf n hx∗n , x − xn i ≥ 0.
We will use the following characterization inspired by [1, Thm 4.5].
Lemma 20 Suppose that X has a Fréchet differentiable bump function and let h : X → R be as in
Proposition 18. Given a valuable subdifferential ∂ on X such that ∂f is contained in ∂ † f, the function f is
approximately convex around x0 if, and only if, there exist ρ, τ > 0 a continuously differentiable function
ϕ : R → R such that ϕ(0) = 0, ϕ0 (0) = 0 and
hx∗ , y − xi ≤ f (y) − f (x) + ϕ (h(x − y))
∀x, y ∈ B(x0 , ρ), ∀x∗ ∈ ∂f (x).
(28)
Proof. Suppose f is approximately convex around x0 . Using Theorem 7 we can find a modulus
µ : R+ → R+ ∪ {+∞} (i.e. a nondecreasing function continuous at 0 with µ(0) = 0) such that
hx∗ , y − xi ≤ f (y) − f (x) + µ (kx − yk) kx − yk
∀x, y ∈ B(x0 , ρ), ∀x∗ ∈ ∂f (x).
It suffices to set for r ∈ R+
−1
µ(r) = sup{kx − yk
[hx∗ , x − yi − f (y) + f (x)] : x, y ∈ B(x0 , r), ∀x∗ ∈ ∂f (x)}.
13
(29)
Let h : X → R be as in Proposition 18. Then we also have
hx∗ , y − xi ≤ f (y) − f (x) + µ (h(x − y)) h(x − y).
(30)
Applying [59, Lemma 3.7], we get some τ > 0 and a continuously differentiable function ϕ : R → R such
that ϕ(0) = 0, ϕ0 (0) = 0 and ϕ(t) ≥ tµ(t) for each t ∈ [0, τ ]. Let ρ := τ /2c, where c is as in Proposition
18. Then for x, y ∈ B(x0 , ρ), x∗ ∈ ∂f (x) we have t := h(y − x) ≤ c ky − xk ≤ τ and
hx∗ , y − xi ≤ f (y) − f (x) + ϕ (h(y − x)) .
Conversely, if there exists ρ > 0 and a differentiable function ϕ : [0, 2ρ] → R satisfying ϕ(0) = 0, ϕ0 (0) = 0
and relation (28) we obtain that relation (29) is satisfied when setting µ(r) := sup{cϕ(s)/s : s ∈ (0, cr]}
for r > 0, µ(0) = 0, what clearly defines a modulus.
We are ready to tackle representation results.
Proposition 21 Let f : X → R ∪ {+∞} be l.s.c and approximately convex around x0 ∈ dom f . Assume
there exists a potential h on X. Then there exist an open neighborhood V of x0 , a set S, a function
g : S × V → R such that, for each s ∈ S, gs is continuously differentiable on V and
f (x) := supg(s, x).
s∈S
Proof. Let ∂ be a valuable subdifferential on X such that ∂f is contained in ∂ † f and let ρ, ϕ be as
in the preceding lemma, so that, for all y ∈ V := B(x0 , ρ), we have
f (y) ≥ sup{f (z) + hz ∗ , y − zi − ϕ(h(y − z)) : z ∈ B(x0 , ρ), z ∗ ∈ ∂f (z)}.
We set
S = {s = (z, z ∗ ) : z ∈ B(x0 , ρ), z ∗ ∈ ∂f (z)},
gs (x) := g(s, x) := f (z) + hz ∗ , x − zi − ϕ(h(x − z))
for (s, x) ∈ S × V.
(xn , x∗n )
in the graph of ∂f such that (xn ) → x and
Then, using Lemma 19, we can pick a sequence
lim inf n hx∗n , x − xn i ≥ 0, and since f is lower semicontinuous, we see that
sup g(s, x) ≥ lim inf g((xn , x∗n ), x) ≥ f (x),
s∈S
hence f (x) = sups∈S g(s, x) for each x ∈ B(x0 , ρ). The continuous differentiability of gs (.) follows from
Lemma 17.
For locally Lipschitzian functions one can get a characterization of approximate convexity in sufficiently
smooth Banach spaces.
Theorem 22 Let f : X → R ∪ {+∞} be locally Lipschitz around x0 ∈ dom f . Suppose there is a potential
h : X → R satisfying conditions (P1 )-(P2 ) in Lemma 17. Then f is approximately convex at x0 if and
only if there exist an open neighborhood V of x0 , a set S and a family (gs )s∈S of differentiable functions
on V such that {gs0 (x0 ) : s ∈ S} is bounded, (gs0 ) is equicontinuous on V and
f (x) := supg(s, x).
s∈S
Proof. The sufficient condition is given in Corollary 15. Let us derive the necessary condition from
the preceding result. Let S and (gs )s∈S be as in its proof. Without loss of generality, we may suppose
x0 = 0. The fact that {gs0 (x0 ) : s ∈ S} is bounded follows from Lemma 17 and the expression of gs0 for
s := (u, u∗ ) :
gs0 (x) = u∗ − (ϕ ◦ h)0 (x − u);
we note that ∂f is bounded around x0 since ∂f is contained in ∂ † f and f is Lipschitz around x0 .
Lemma 17 shows that ϕ ◦ h is uniformly differentiable on B(x0 , ρ). It follows from Lemma 16 that the
family (gs0 )s∈S is equicontinuous on B(x0 , ρ).
14
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