Journal of Computer Science and Cybernetics, Vol.22, No.3 (2006), 235—243
FINITE-DIMENSIONAL APPROXIMATION FOR ILL-POSED VECTOR
OPTIMIZATION OF CONVEX FUNCTIONALS IN BANACH SPACES
NGUYEN THI THU THUY
1
, NGUYEN BUONG
2
1
Faculty of Sciences, Thai Nguyen University
2
Institute of Information Technology
Abstract. In this paper we present the convergence and convergence rate for regularization solutions
in connection with the finite-dimensional approximation for ill-posed vector optimization of convex
functionals in reflexive Banach space. Convergence rates of its regularized solutions are obtained on
the base of choosing the regularization parameter a priory as well as a posteriori by the modified
generalized discrepancy principle. Finally, an application of these results for convex optimization
problem with inequality constraints is shown.
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c.
1. INTRODUCTION
Let
X
be a real reflexive Banach space preserved a property that
X
and
X
∗
are strictly
convex, and weak convergence and convergence of norms of any sequence in
X
imply its strong
convergence, where
X
∗
denotes the dual space of
X
. For the sake of simplicity, the norms
of
X
and
X
∗
are denoted by the symbol
.
. The symbol
x
∗
, x
denotes the value of the
linear continuous functional
x
∗
∈ X
∗
at the point
x ∈ X
. Let
ϕ
j
(x)
,
j = 0, 1, , N
, be the
weakly lower semicontinuous proper convex functionals on
X
that are assumed to be Gˆateaux
differentiable with the hemicontinuous derivatives
A
j
(x)
at
x ∈ X
.
In [6], one of the authors has considered a problem of vector optimization: find an element
u ∈ X
such that
ϕ
j
(u) = inf
x∈X
ϕ
j
(x), ∀j = 0, 1, , N. (1.1)
Set
Q
j
=
ˆx ∈ X : ϕ
j
(ˆx) = inf
x∈X
ϕ
j
(x)
, j = 0, 1, , N, Q =
N
j=0
Q
j
.
It is well know that
Q
j
coincides with the set of solutions of the following operator equation
A
j
(x) = θ, (1.2)
and is a closed convex subset in
X
(see [11]). We suppose that
Q = ∅
, and
θ /∈ Q
, where
θ
is
the zero element of
X
(or
X
∗
).
236
NGUYEN THI THU THUY, NGUYEN BUONG
In [6] it is showed the existence and uniqueness of the solution
x
h
α
of the operator equation
N
j=0
α
λ
j
A
h
j
(x) + αU(x) = θ, (1.3)
λ
0
= 0 < λ
j
< λ
j+1
< 1, j = 1, 2, , N − 1,
where
α > 0
is the small parameter of regularization,
U
is the normalized duality mapping of
X
, i.e.,
U : X → X
∗
satisfies the condition
U(x), x = x
2
, U(x) = x,
A
h
j
are the hemicontinuous monotone approximations for
A
j
in the forms
A
j
(x) − A
h
j
(x) hg(x), ∀x ∈ X, (1.4)
with level
h → 0
, and
g(t)
is a bounded (the image of the bounded set is bounded) nonnegative
function,
t 0
.
Clairly, the convergence and convergence rates of the sequence
x
h
α
to
u
depend on the
choice of
α = α(h)
. In [6], one has showed that the parameter
α
can be chosen by the
modified generalized discrepancy principle, i.e.,
α = α(h)
is constructed on the basis of the
following equation
ρ(α) = h
p
α
−q
, p, q > 0, (1.5)
where
ρ(α) = α(a
0
+ t(α))
, the function
t(α) = x
h
α
depends continuously on
α α
0
> 0
,
a
0
is some positive constant.
In computation the finite-demensional approximation for (1.3) is the important problem.
As usualy, it can be aproximated by the following equation
N
j=0
α
λ
j
A
hn
j
(x) + αU
n
(x) = θ, x ∈ X
n
, (1.6)
where
A
hn
j
= P
∗
n
A
h
j
P
n
, U
n
= P
∗
n
UP
n
and
P
n
: X −→ X
n
the linear projection from
X
onto
X
n
,
X
n
is the finite-dimensional subspace of
X
,
P
∗
n
is the conjugate of
P
n
,
X
n
⊂ X
n+1
, ∀n, P
n
x −→ x, ∀x ∈ X.
Without loss of generality, suppose that
P
n
= 1
(see [11]).
As for (1.3), equation (1.6) has also an unique solution
x
h
α,n
, and for every fixed
α > 0
the
sequence
{x
h
α,n
}
converges to
x
h
α
, the solution of (1.3), as
n → ∞
(see [11]).
The natural problem is to ask whether the sequence
{x
h
α,n
}
converges to
u
as
α, h → 0
and
n → ∞
, and how fast it converges, where
u
is an element in
Q
. The purpose of this paper is
to answer these questions.
We assume, in addition, that
U
satisfies the condition
U(x) − U(y), x − y m
U
x − y
s
, m
U
> 0, s 2, ∀x, y ∈ X. (1.7)
Set
γ
n
(x) = (I − P
n
)x, x ∈ Q,
where
I
denotes the identity operator in
X
.
FINITE-DIMENSIONAL APPROXIMATION FOR ILL-POSED VECTOR OPTIMIZATION
237
Hereafter the symbols
and
→
indicate weak convergence and convergence in norm,
respectively, while the notation
a ∼ b
is meant
a = O(b)
and
b = O(a).
2. MAIN RESULT
The convergence of
{x
h
α,n
}
to
u
is determined by the following theorem.
Theorem 1. If
h/α
and
γ
n
(x)/α → 0
, as
α → 0
and
n → ∞
, then the sequence
x
h
α,n
converges to
u
.
Proof. For
x ∈ Q, x
n
= P
n
x
, it follows from (1.6) that
N
j=0
α
λ
j
A
hn
j
(x
h
α,n
), x
h
α,n
− x
n
+ αU
n
(x
h
α,n
) − U
n
(x
n
), x
h
α,n
− x
n
= αU
n
(x
n
), x
n
− x
h
α,n
.
Therefore, on the basis of (1.2), (1.7) and the monotonicity of
A
hn
j
= P
∗
n
A
h
j
P
n
, and
P
n
P
n
= P
n
we have
αm
U
x
h
α,n
− x
n
s
αU (x
h
α,n
) − U(x
n
), x
h
α,n
− x
n
= αU
n
(x
h
α,n
) − U
n
(x
n
), x
h
α,n
− x
n
=
N
j=0
α
λ
j
A
hn
j
(x
h
α,n
), x
n
− x
h
α,n
+ αU
n
(x
n
), x
n
− x
h
α,n
N
j=0
α
λ
j
A
hn
j
(x
n
), x
n
− x
h
α,n
+ αU
n
(x
n
), x
n
− x
h
α,n
=
N
j=0
α
λ
j
A
h
j
(x
n
) − A
j
(x
n
) + A
j
(x
n
) − A
j
(x), x
n
− x
h
α,n
+ αU(x
n
), x
n
− x
h
α,n
. (2.1)
On the other hand, by using (1.4) and
A
j
(x
n
) − A
j
(x) Kγ
n
(x),
where
K
is some positive constant depending only on
x
, it follows from (2.1) that
m
U
x
h
α,n
− x
n
s
1
α
(N + 1)
hg(x
n
) + Kγ
n
(x)
x
n
− x
h
α,n
+ U(x
n
), x
n
− x
h
α,n
. (2.2)
Because of
h/α
,
γ
n
(x)/α → 0
as
α → 0
,
n → ∞
and
s 2
, this inequality gives us the
boundedness of the sequence
{x
h
α,n
}
. Then, there exists a subsequence of the sequence
{x
h
α,n
}
converging weakly to
ˆx
in
X
. Without loss of generality, we assume that
x
h
α,n
ˆx
as
h, h/α → 0
and
n → ∞
. First, we prove that
ˆx ∈ Q
0
. Indeed, by virtue of the monotonicity
of
A
hn
j
= P
∗
n
A
h
j
P
n
,
U
n
= P
∗
n
UP
n
and (1.6) we have
A
hn
0
(P
n
x), P
n
x − x
h
α,n
A
hn
0
(x
h
α,n
), P
n
x − x
h
α,n
=
N
j=1
α
λ
j
A
hn
j
(x
h
α,n
), x
h
α,n
− P
n
x + αU
n
(x
h
α,n
), x
h
α,n
− P
n
x
N
j=1
α
λ
j
A
hn
j
(P
n
x), x
h
α,n
− P
n
x + αU
n
(P
n
x), x
h
α,n
− P
n
x, ∀x ∈ X.
238
NGUYEN THI THU THUY, NGUYEN BUONG
Because of
P
n
P
n
= P
n
, so the last inequality has form
A
h
0
(P
n
x), P
n
x − x
h
α,n
N
j=1
α
λ
j
A
h
j
(P
n
x), x
h
α,n
− P
n
x + αU (P
n
x), x
h
α,n
− P
n
x, ∀x ∈ X.
By letting
h, α → 0
and
n → ∞
in this inequality we obtain
A
0
(x), x − ˆx 0, ∀x ∈ X.
Consequently,
ˆx ∈ Q
0
(see [11]). Now, we shall prove that
ˆx ∈ Q
j
,
j = 1, 2, , N
. Indeed, by
(1.6) and making use of the monotonicity of
A
hn
j
and
U
n
, it follows that
α
λ
1
A
hn
1
(x
h
α,n
),x
h
α,n
− P
n
x +
N
j=2
α
λ
j
A
hn
j
(x
h
α,n
), x
h
α,n
− P
n
x + αU
n
(x
h
α,n
), x
h
α,n
− P
n
x
= α
λ
0
A
hn
0
(x
h
α,n
), P
n
x − x
h
α,n
A
hn
0
(P
n
x), P
n
x − x
h
α,n
= A
h
0
(P
n
x) − A
0
(P
n
x) + A
0
(P
n
x) − A
0
(x), P
n
x − x
h
α,n
, ∀x ∈ Q
0
.
Therefore,
A
h
1
(P
n
x), x
h
α,n
− P
n
x +
N
j=2
α
λ
j
−λ
1
A
h
j
(P
n
x), x
h
α,n
− P
n
x + α
1−λ
1
U(P
n
x), x
h
α,n
− P
n
x
1
α
hα
1−λ
1
g(P
n
x) + Kγ
n
(x)
P
n
x − x
h
α,n
, ∀x ∈ Q
0
.
After passing
h, α → 0
and
n → ∞
, we obtain
A
1
(x), ˆx − x 0, ∀x ∈ Q
0
.
Thus,
ˆx
is a local minimizer for
ϕ
1
on
S
0
(see [9]). Since
S
0
∩ S
1
= ∅
, then
ˆx
is also a global
minimizer for
ϕ
1
, i.e.,
ˆx ∈ S
1
.
Set
˜
Q
i
= ∩
i
k=0
Q
k
. Then,
˜
Q
i
is also closed convex, and
˜
Q
i
= ∅
.
Now, suppose that we have proved
ˆx ∈
˜
Q
i
and we need to show that
ˆx
belongs to
Q
i+1
.
Again, by virtue of (1.6) for
x ∈
˜
Q
i
, we can write
A
hn
i+1
(x
h
α,n
), x
h
α,n
− P
n
x +
N
j=i+2
α
λ
j
−λ
i+1
A
hn
j
(x
h
α,n
), x
h
α,n
− P
n
x
+ α
1−λ
i+1
U
n
(x
h
α,n
), x
h
α,n
− P
n
x =
i
k=0
α
λ
k
−λ
i+1
A
hn
k
(x
h
α,n
), P
n
x − x
h
α,n
1
α
i
k=0
α
λ
k
+1−λ
i+1
A
h
k
(P
n
x) − A
k
(P
n
x) + A
k
(P
n
x) − A
k
(x), P
n
x − x
h
α,n
1
α
(i + 1)
hg(P
n
x) + Kγ
n
(x)
P
n
(x) − x
h
α,n
.
Therefore,
FINITE-DIMENSIONAL APPROXIMATION FOR ILL-POSED VECTOR OPTIMIZATION
239
A
h
i+1
(P
n
x), x
h
α,n
− P
n
x +
N
j=i+2
α
λ
j
−λ
i+1
A
h
j
(P
n
x), x
h
α,n
− P
n
x
+α
1−λ
i+1
U(P
n
x), x
h
α,n
− P
n
x
hg(P
n
x) + Kγ
n
(x)
α
(N + 1)P
n
x − x
h
α,n
.
By letting
h, α → 0
and
n → ∞
, we have
A
i+1
(x), ˆx − x 0, ∀x ∈
˜
Q
i
.
As a result,
ˆx ∈ Q
i+1
.
On the other hand, it follows from (2.2) that
U(x), x − ˆx 0, ∀x ∈ Q.
Since
Q
j
is closed convex,
Q
is also closed convex. Replacing
x
by
tˆx + (1 − t)x
,
t ∈ (0, 1)
in
the last inequality, and dividing by
(1 − t)
and letting
t
to
1
, we obtain
U(ˆx), x − ˆx 0, ∀x ∈ Q.
Hence
ˆx x
,
∀x ∈ Q
. Because of the convexity and the closedness of
Q
, and the strictly
convexity of
X
we deduce that
ˆx = u
. So, all sequence
{x
h
α,n
}
converges weakly to
u
. Set
x
n
= u
n
= P
n
u
in (2.2) we deduce that the sequence
{x
h
α,n
}
converges strongly to
u
as
h → 0
and
n → ∞
. The proof is complete.
In the following, we consider the finite-dimensional variant of the generalized discrepancy
principle for the choice
˜α = α(h, n)
so that
x
h
˜α,n
converges to
u
, as
h, α → 0
and
n → ∞
.
Note that, the generalized discrepancy principle for parameter choice is presented first in
[8] for the linear ill-posed problems. For the nonlinear ill-posed equation involving a monotone
operator in Banach space the use of a discrepancy principle to estimate the rate of convergence
of the regularized solutions was considered in [5]. In [4] the convergence rates of regularized
solutions of ill-posed variational inequalities under arbitrary perturbative operators were in-
vestigated when the regularization parameter was chosen arbitrarily such that
α ∼ (δ + ε)
p
,
0 < p < 1
. In this paper, we consider the modified generalized discrepancy principle for
selecting
˜α
in connection with the finite-dimensional and obtain the rates of convergence for
the regularized solutions in this case.
The parameter
α(h, n)
can be chosen by
α(a
0
+ x
h
α,n
) = h
p
α
−q
, p, q > 0 (2.3)
for each
h > 0
and
n
. It is not difficult to verify that
ρ
n
(α) = α(a
0
+ x
h
α,n
)
possesses all
properties as well as
ρ(α)
does, and
lim
α→+∞
α
q
ρ
n
(α) = +∞, lim
α→+0
α
q
ρ
n
(α) = 0.
To find
α
by (2.3) is very complex. So, we consider the following rule.
The rule. Choose
˜α = α(h, n) α
0
:= (c
1
h + c
2
γ
n
)
p
,
c
i
> 1, i = 1, 2
,
0 < p < 1
such that
the following inequalities
˜α
1+q
(a
0
+ x
h
˜α,n
) d
1
h
p
,
˜α
1+q
(a
0
+ x
h
˜α,n
) d
2
h
p
, d
2
d
1
> 1,
240
NGUYEN THI THU THUY, NGUYEN BUONG
hold.
In addition, assume that
U
satisfies the following condition
U(x) − U(y) C(R)x − y
ν
, 0 < ν 1, (2.4)
where
C(R), R > 0
, is a positive increasing function on
R = max{x, y}
(see [10]).
Set
γ
n
= max
x∈Q
{γ
n
(x)}.
Lemma 1.
lim
h→0
n→∞
α(h, n) = 0.
Proof. Obviously, it follows from the rule that
α(h, n) d
1/(1+q)
2
a
0
+ x
h
α(h,n),n
−1/(1+q)
h
p/(1+q)
d
1/(q+1)
2
a
−1/(1+q)
0
h
p/(1+q)
.
Lemma 2. If
0 < p < 1
then
lim
h→0
n→∞
h + γ
n
α(h, n)
= 0.
Proof. Obviously using the rule we get
h + γ
n
α(h, n)
c
1
h + c
2
γ
n
(c
1
h + c
2
γ
n
)
p
= (c
1
h + c
2
γ
n
)
1−p
→ 0
as
h → 0
and
n → ∞
.
Now, let
x
h
˜α,n
be the solution of (1.6) with
α = ˜α
. By the argument in the proof of
Theorem 1, we obtain the following result.
Theorem 2. The sequence
x
h
˜α,n
converges to
u
as
h → 0
and
n → ∞
.
The next theorem shows the convergence rates of
{x
h
˜α,n
}
to
u
as
h → 0
and
n → ∞
.
Theorem 3. Assume that the following conditions hold:
(i)
A
0
is continuously Frchet differentiable, and satifies the condition
A
0
(x) − A
0
(u)(x − u) τA
0
(x), ∀u ∈ Q,
where
τ
is a positive constant, and
x
belongs to some neighbourhood of
Q
;
(ii)
A
h
(X
n
)
are contained in
X
∗
n
for sufficiently large
n
and small
h
;
(iii)
there exists an element
z ∈ X
such that
A
0
(u)
∗
z = U(u)
;
(vi)
the parameter
˜α = α(h, n)
is chosen by the rule.
Then, we have
x
h
˜α,n
− u = O
(h + γ
n
)
η
1
+ γ
η
2
n
,
η
1
= min
1 − p
s − 1
,
µ
1
p
s(1 + q)
, η
2
= min
1
s
,
ν
s − 1
.
FINITE-DIMENSIONAL APPROXIMATION FOR ILL-POSED VECTOR OPTIMIZATION
241
Proof. Replacing
x
n
by
u
n
= P
n
u
in
(2.2)
we obtain
m
U
x
h
˜α,n
− u
n
s
1
˜α
(N + 1)hg(u
n
) + Kγ
n
u
n
− x
h
˜α,n
+U(u
n
) + U(u) − U(u), u
n
− x
h
˜α,n
. (2.5)
By
(2.4)
it follows that
U(u
n
) − U(u), u
n
− x
h
˜α,n
C(
˜
R)u
n
− u
ν
u
n
− x
h
˜α,n
C(
˜
R)γ
ν
n
u
n
− x
h
˜α,n
, (2.6)
where
˜
R > u.
On the other hand, using conditions
(i), (ii), (iii)
of the theorem we can write
U(u), u
n
− x
h
˜α,n
= U(u), u
n
− u + z, A
0
(u)(u − x
h
˜α,n
)
˜
Rγ
n
+ z(τ + 1)A
0
(x
h
˜α,n
)
˜
Rγ
n
+ z(τ + 1)
hg(x
h
˜α,n
) + A
h
0
(x
h
˜α,n
)
˜
Rγ
n
+ z(τ + 1)
N
j=1
˜α
λ
j
A
h
j
(x
h
˜α,n
) + ˜αx
h
˜α,n
+ hg(x
h
˜α,n
)
. (2.7)
Combining (2.6) and (2.7) inequality (2.5) has form
m
U
x
h
˜α,n
− u
n
s
1
˜α
(N + 1)hg(u
n
) + Kγ
n
u
n
− x
h
˜α,n
+ C(
˜
R)γ
ν
n
u
n
− x
h
˜α,n
+
˜
Rγ
n
+ z(τ + 1)
N
j=1
˜α
λ
j
A
h
j
(x
h
˜α,n
) + ˜αx
h
˜α,n
+ hg(x
h
˜α,n
)
. (2.8)
On the other hand, making use of the rule and the boundedness of
{x
h
˜α,n
}
it implies that
˜α = α(h, n) (c
1
h + c
2
γ
n
)
p
,
˜α = α(h, n) C
1
h
p/(1+q)
, C
1
> 0,
˜α = α(h, n) 1,
for sufficiently small
h
and large
n
.
Consequently, in view of (2.8) it follows that
m
U
x
h
˜α,n
− u
n
(N + 1)hg(u
n
) + Kγ
n
(c
1
h + c
2
γ
n
)
p
+ C(
˜
R)γ
ν
n
u
n
− x
h
˜α,n
+
˜
Rγ
n
+ C
2
(h + γ
n
)
λ
1
p/(1+q)
˜
C
1
(h + γ
n
)
1−p
+ γ
ν
n
u
n
− x
h
˜α,n
+
˜
C
2
γ
n
+
˜
C
3
(h + γ
n
)
λ
1
p/(1+q)
,
C
2
and
˜
C
i
,
i = 1, 2, 3
are the positive constants.
242
NGUYEN THI THU THUY, NGUYEN BUONG
Using the implication
a, b, c 0, p
1
> q
1
, a
p
1
ba
q
1
+ c ⇒ a
p
1
= O
b
p
1
/(p
1
−q
1
)
+ c
we obtain
x
h
˜α,n
− u
n
= O
(h + γ
n
)
η
1
+ γ
η
2
n
.
Thus,
x
h
˜α,n
− u = O
(h + γ
n
)
η
1
+ γ
η
2
n
,
which completes the proof.
Remarks. If
˜α = α(h, n)
is chosen a priori such that
˜α ∼ (h + γ
n
)
η
,
0 < η < 1
, then
inequality (2.8) has the form
m
U
x
h
˜α,n
− u
n
C
1
(h + γ
n
)
1−η
+ γ
ν
n
u
n
− x
h
˜α,n
+ C
2
γ
n
+ C
3
(h + γ
n
)
λ
1
η
,
where
C
i
, i = 1, 2, 3
are the positive constants.
Therefore,
x
h
˜α,n
− u
n
= O
(h + γ
n
)
θ
1
+ γ
θ
2
n
,
whence,
x
h
˜α,n
− u = O
(h + γ
n
)
θ
1
+ γ
θ
2
n
,
θ
1
= min
1 − η
s − 1
,
λ
1
η
s
, θ
2
= min
1
s
,
ν
s − 1
.
3. AN APPLICATION
In this section we consider a constrained optimization problem:
inf
x∈X
f
N
(x) (3.1)
subject to
f
j
(x) 0, j = 0, , N − 1, (3.2)
where
f
0
, f
1
, , f
N
are weakly lower semicontinuous and properly convex functionals on
X
that are assumed to be Gteaux differentiable at
x ∈ X
.
Set
Q
j
= {x ∈ X : f
j
(x) 0}, j = 0, , N − 1. (3.3)
Obviously,
Q
j
is the closed convex subset of
X
,
j = 0, , N − 1
.
Define
ϕ
N
(x) = f
N
(x), ϕ
j
(x) = max{0, f
j
(x)}, j = 0, , N − 1. (3.4)
Evidently,
ϕ
j
are also convex functionals on
X
and
Q
j
= {¯x ∈ X : ϕ
j
(¯x) = inf
x∈X
ϕ
j
(x)}, 0, 1, , N.
So,
¯x
is a solution of the problem:
ϕ
j
(¯x) = inf
x∈X
ϕ
j
(x), ∀j = 0, 1, , N.
FINITE-DIMENSIONAL APPROXIMATION FOR ILL-POSED VECTOR OPTIMIZATION
243
REFERENCES
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discontinuous operators, Differ. Uravn. 25 (1979) 331—342.
[3] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff
Int. Publ. Leyden (Ed. Acad. Bucuresti, Romania, Netherlands) 1976.
[4] Ng. Buong, Convergence rates and finite-dimensional approximation for a class of ill-posed
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[7] I. Ekeland and R. Temam, Convex analysis and variational problems, North-Holland Publ.
Company, Amsterdam, Holland, 1970.
[8] H. W. Engl, Discrepancy principle for tikhonov regularization of ill-posed problems lead-
ing to optimal convergence rates, J. of Optimization Theory and Appl. 52 (1987) 209—215.
[9] I. P. Ryazantseva, Operator method of ragularization for problems of optimal program-
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Received on May 29, 2006
Revised on August 2, 2006
. phiˆe ´ m h`am lˆo ` i trong khˆong gian Banach pha ’ n xa . . Tˆo ´ c dˆo . hˆo . i tu . cu ’ a nghiˆe . m hiˆe . u chı ’ nh nhˆa . n du . o . . c du . . a trˆen viˆe . c cho . n tham sˆo ´ hiˆe . u chı ’ nh. t˘a ´ t. Trong b`ai b´ao n`ay ch´ung tˆoi tr`ınh b`ay su . . hˆo . i tu . v`a tˆo ´ c dˆo . hˆo . i tu . cu ’ a nghiˆe . m hiˆe . u chı ’ nh trong xˆa ´ p xı ’ h˜u . u ha . n chiˆe ` u cho b`ai. vector optimization of convex functionals in reflexive Banach space. Convergence rates of its regularized solutions are obtained on the base of choosing the regularization parameter a priory as well