2.3 Some geometric properties of Banach spaces
2.3.5 Basic notions of convex analysis
We give the following important definitions which are used in subsequent chapters.
Let E be a real Banach space and f : E → (−∞,+∞] be a proper convex and lower semicontinuous function. We denote the domain of f byDomf ={x∈E :f(x)<+∞}.
Definition 2.3.27. LetD be a convex subset of a vector space X and f :D→R∪ {+∞}
be a map. Then,
(i) f is convex if for each λ∈[0,1] and x, y ∈D, we have
f(λx+ (1−λ)y)≤λf(x) + (1−λ)f(y);
(ii) f is called proper if there exists at least one x∈D such that f(x)̸= +∞;
(iii) f :Dom(f)→(−∞,∞] is lower semi-continuous at a point x∈Dom(f) if f(x)≤lim inf
n→∞ f(xn), (2.10)
for each sequence {xn} in Dom(f) such that lim
n→∞xn = x. An example of a convex and lower semicontinuous function is the indicator function δC :H →R of a nonempty closed and convex subset C of H defined by
δC(x) =
(0, if x∈C, +∞, otherwise;
(iv) f is upper semi-continuous at x0 ∈D if f(x0)≥lim sup
x→x0
f(x).
Definition 2.3.28. Let H be a real Hilbert space and A : H → H be a bounded linear map. We define a map A∗ :H →H by the relation
⟨Ax, y⟩=⟨x, A∗y⟩,
for all x, y ∈H. The map A∗ is called the adjoint/dual of A.
Let x∈int domf, the subdifferential off atx is the convex set defined by
∂f(x) ={x∗ ∈E∗ :f(x) +⟨y−x, x∗⟩ ≤f(y), ∀ x∈E}.
The F´enchel conjugate off is the convex function f∗ :E∗ →(−∞,+∞] defined by f∗(x∗) = sup{⟨x∗, x⟩ −f(x) :x∈E}.
It is known that f satisfies the Young-F´enchel inequality
⟨x∗, x⟩ ≤f(x) +f∗(x∗) x∈E, x∗ ∈E∗.
Moreover, the inequality holds if x∗ ∈∂f(x).
Given x∈ int domf and y ∈ E, the right-hand derivative of f at x in the direction of y is defined by
f0(x, y) := lim
t→0+
f(x+ty)−f(x)
t . (2.11)
The function f is said to be Gˆateaux differentiable at x if (2.11) exists for any y. In this case, the gradient of f at x is the linear function ▽f(x) defined by ⟨y,▽f(x)⟩ :=
f0(x, y) for ally ∈E. The functionf is said to be Gˆateaux differentiable if it is Gˆateaux differentiable at each x ∈ int domf. When the limit as t → 0 in (2.11) is attained uniformly for any y ∈ E with ||y|| = 1, we say that f is uniformly Fr´echet differentiable at x. Also, if f is Fr´echet differentiable, the ▽f is norm to norm continuous (see [141]).
Definition 2.3.29. The function f :E →(−∞,+∞] is called Legendre if it satisfies the following conditions:
1. f is Gˆateaux differentiable, int domf ̸=∅ and dom▽f =int domf, 2. f∗ is Gˆateaux differentiable, int domf∗ ̸=∅ and dom▽f∗ =int domf∗.
One important and interesting Legendre function is 1p||.||p (1< p <∞), where the Banach space E is smooth and strictly convex, and in particular the Hilbert space. For more examples of Legendre function, (see [22,24]).
Remark 2.3.30. If E is a real reflexive Banach space andf is a Legendre function, then we have
1. f is a Legendre function if and only if f∗ is a Legendre function, 2. (∂f)−1 =∂f∗,
3. ▽f = (▽f∗)−1, ran ▽f = dom ▽ f∗ = int(domf∗), ran ▽ f∗ = dom ▽f = int(domf);
4. f and f∗ are strictly convex on the interior of their respective domains.
Remark 2.3.31. If f :E →R is Gˆateaux differentiable and convex, then
⟨y,▽f(x)⟩=f0(x, y) = lim
t→0
f(x+ty)−f(x) t
= lim
t→0
f((1−t)x+t(x+y))−f(x) t
≤lim
t→0
(1−t)f(x) +tf(x+y)−f(x) t
=f(x+y)−f(x).
Definition 2.3.32. [30] Let f :E →R be a convex and Gˆateaux differentiable function, the Bregman distance with respect to f is the bifunction Df :domf ×int domf →[0,∞) defined by
Df(x, y) = f(x)−f(y)− ⟨x−y,▽f(y)⟩. (2.12) It is worth mentioning thatDf is not a distance in the usual sense but enjoys the following properties:
1. Df(x, x) = 0, but Df(x, y) = 0 may not simply imply x=y, 2. Df is not symmetric and does not satisfy the triangle inequality, 3. for x∈domf and y, z ∈domf, we have
Df(x, y) +Df(y, z)−Df(x, z)≤ ⟨▽f(z)− ▽f(y), x−y⟩, (2.13) 4. for each z ∈ E, we have Df(z,▽f∗(PN
i=1ti ▽ f(xi))) ≤ PN
i=1tiDf(z, xi), where {xi}Ni=1 ⊆E and {ti}Ni=1 ⊆(0,1) satisfies PN
i=1ti = 1.
More so, it is well known that the duality mapping JpE is the sub-differential of the func- tional fp(.) = 1p||.||p for p >1, (see [61]). Then, the Bregman distance Dp is defined with respect to fp as follows:
Dp(x, y) = 1
p||y||p−1
p||x||p− ⟨JPEx, y−x⟩
= 1
q||x||p− ⟨JpEx, y⟩+ 1 p||y||p
= 1
q||x||p− 1
q||y||p− ⟨JpEx−JpEy, y⟩. (2.14) Bregman distance has been studied by many researchers because of its nice and effective characteristics in analyzing optimization and feasibility algorithms, (see [24, 38, 39, 40]
and the references contained in). The function Vf :E×E∗ →[0,+∞] associated with f, which is defined by [41]
Vf(x, x∗) =f(x)− ⟨x, x∗⟩+f∗(x∗), ∀ x∈E. (2.15) Then Vf(x, x∗) = Df(x,▽f∗(x∗)) for all x∈E and x∗ ∈E∗. Moreover, by subdifferential inequality, we have
Vf(x, x∗) +⟨▽f∗(x∗)−x, y∗⟩ ≤Vf(x, x∗+y∗), for all x∈E and x∗, y∗ ∈E∗.
The modulus of total convexity at xis the bifunctionυf :int domf×[0,+∞)→[0,+∞) defined by
υf(x, t) :=int{Df(y, x) :y ∈domf,||y−x||=t}.
The function f is said to be totally convex at x∈ int domf if υf(x, t) is positive for any t >0. LetC be a nonempty subset of E, the modulus of total convexity of f onC is the bifunction υf :int domf ×[0,+∞)→[0,+∞) defined by
υf(C, t) :={υf(x, t) :x∈C∩int domf}.
The function f is called totally convex on bounded subsets if υf(C, t) is positive for any nonempty and bounded subset C and any t >0.
Proposition 2.3.33. [197] Ifx∈int domf, then the following statements are equivalent:
1. the function f is totally convex, 2. for any sequence {yn} ⊂domf,
n→∞lim Df(yn, x) = 0 =⇒ lim
n→∞∥yn−x∥= 0.
Definition 2.3.34. [39] A function f is called sequentially consistent if for any two se- quences {xn} and {yn} in E such that the first one is bounded,
n→∞lim Df(yn, xn) = 0 =⇒ lim
n→∞∥yn−xn∥= 0.
Proposition 2.3.35. [41] If domf contains at least two points, then the function f is totally convex on bounded sets if and only if the function f is sequentially consistent.
Proposition 2.3.36. [194] Let f :E →R be a Gˆateaux differentiable and totally convex function. If x0 ∈ E and the sequence {Df(xn, x0)} is bounded, then the sequence {xn} is bounded.
The Bregman projection [30] (P rojCf(x)) with respect to f at x ∈ int(domf) onto a nonempty, closed and convex set C⊂int(domf) is defined by
Df(P rojCf(x), x) = inf
y∈CDf(y, x). (2.16)
It is well-known that (see [39])z =P rojCfx if and only if ⟨▽f(x)− ▽f(z), y−z⟩ ≤0 for all y∈C. We also have
Df(y, P rojCf(x)) +Df(P rojCf(x), x)≤Df(y, x), ∀x∈E, y ∈C.
Similar to the metric projection in Hilbert spaces, the Bregman projection with respect to totally and Gˆateaux differentiable functions has a variational inequality characterization, (see [39]). Using (2.15), the function Vp :E×E∗ →[0,+∞) for 2≤p <∞ is defined by
Vp(x, y) = 1
q||x||p− ⟨x, y⟩+ 1
p||y||p, ∀ x∗ ∈E∗, x∈E.
Vp is nonnegative and Vp(x∗, x) =Dp(JqE∗(x∗), x) for allx∈E∗ and y∈E∗. Moreover, by the subdifferential inequality
⟨▽f(x), y−x⟩ ≤f(y)−f(x),
with f(x) = 1q∥x∥q, x∈E∗, then▽f(x) =JqE∗(x). So, we have
⟨JqE∗(x), y⟩ ≤ 1
q∥x+y∥q− 1
q∥x∥q. (2.17)
From (2.17), we get
Vp(x∗+y∗, x) = 1
q∥x∗+y∗∥ − ⟨x∗+y∗, x⟩+1 q∥x∥p
≥ 1
q||x∗||q+⟨y∗, JqE∗(x∗)⟩ − ⟨x∗+y∗, x⟩+1 q||x||p
= 1
q||x∗||q− ⟨x∗, x⟩+1
p||x||p+⟨y∗, JqE∗(x∗)⟩ − ⟨y∗, x⟩
= 1
q||x∗||q− ⟨x∗, x⟩+1
p||x||p+⟨y∗, JqE∗(x∗)−x⟩
=Vp(x∗, x) +⟨y∗, JqE∗(x∗)−x⟩, for all x∈E and x∗, y∗ ∈E∗.
Forp-uniformly convex space, The Bregman distance also possess the following important properties
Dp(x, y) =Dp(x, z) +Dp(z, y) +⟨z−y, JpEx−JpEy⟩, ∀ x, y, z ∈E.
Dp(x, y) +Dp(y, x) = ⟨x−y, JpEx−JpEy⟩, ∀ x, y ∈E.
It is also known that the norm metric and the Bregman distance has the following relation, (see [204]).
τ||x−y||p ≤Dp(x, y)≤ ⟨x−y, JpEx−JpEy⟩, (2.18) where τ >0 is some fixed number. Let C be a nonempty, closed and convex subset of E, the metric projection defined as
PCx:=argminy∈C||x−y||, x∈E
is the unique minimizer of the norm distance, which can be characterized by a variational inequality:
⟨JpE(x−PCx), z−PCx⟩ ≤0, ∀z ∈C.
Similar to the metric projection is the Bregman projection (the minimizer of the Bregman distance) which is defined as
ΠCx:=argminy∈CDp(x, y), x∈E.
The Bregman projection can also be characterized by a variational inequality:
⟨JpE(x)−JpE(ΠCx), z−ΠCx⟩ ≤0,∀ z ∈C, (2.19) from which one get
Dp(ΠCx, z)≤Dp(x, z)−Dp(x,ΠCx), ∀z ∈C.
Definition 2.3.37. A function f is said to be 1. strongly coercive, if
lim
||xn||→∞
f(xn)
||xn|| =∞.
2. super coercive, if
x→∞lim f(x)
||x|| = +∞.
Definition 2.3.38. [196] A point u ∈ C is said to be an asymptotic fixed point of T : C → C if there exists a sequence {xn} in C such that xn ⇀ u and ||xn−T xn|| → 0. We denote the asymptotic fixed point set of T by Fˆ(T).
Definition 2.3.39. Let C be a nonempty subset of int domf. An operator T : C → int domf is said to be
1. Bregman firmly nonexpansive (BFNE), if
⟨T x−T y,▽f(T x)− ▽f(T y)⟩ ≤ ⟨T x−T y,▽f(x)− ▽f(y)⟩, for any x, y ∈C, or equivalently
Df(T x, T y) +Df(T y, T x) +Df(T x, x) +Df(T y, y)≤Df(T x, y) +Df(T y, x).
2. Bregman quasi firmly nonexpansive (BQFNE), if F(T)̸=∅ and
⟨T x−p,▽f(x)− ▽f(T x)⟩ ≥0, ∀ x∈C, p∈F(T), or equivalently,
Df(p, T x) +Df(T x, x)≤Df(p, x).
3. Bregman quasi-nonexpansive (BQNE), if F(T)̸=∅ and Df(p, T x)≤Df(p, x), x ∈C, p∈F(T).
4. Bregman relatively nonexpansive, if F(T)̸=∅, and
Df(p, T x)≤Df(p, x), ∀ p∈F(T), x∈C and Fˆ(T) =F(T).
5. Bregman Strongly Nonexpansive (BSNE) with Fˆ(T)̸=∅, if Df(p, T x)≤Df(p, x) ∀ x∈C, p∈Fˆ(T) and for any bounded sequence {xn}n≥1 ⊂C,
n→∞lim(Df(p, xn)−Df(p, T xn)) = 0 implies that lim
n→∞Df(T xn, xn) = 0.