RESEARCH ARTICLE An ergodic approach to the Ky Fan inequality over the fixed point set

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Optimization
Vol. 00, No. 00, December 2014, 1–7

RESEARCH ARTICLE
An ergodic approach to the Ky Fan inequality over the fixed point
set
Pham Ngoc Anha
Department of Scientific Fundamentals, Posts and Telecommunications Institute of
Technology, Hanoi, Vietnam.
Trinh N. Hai
School of Applied Mathematics and Informatics, Ha Noi University of Science and
Technology, Vietnam.
(Received 00 Month 200x; in final form 00 Month 200x)
We introduce a new approach for solving the Ky Fan inequality over the fixed point set of a
nonexpansive mapping, where the cost bifunction is monotone and not necessarily Lipschitztype continuous. The proposed algorithms are quite simple and based on the idea of the
ergodic iteration methods. By choosing suitable regularization parameters, we also present
the convergence analysis for the algorithms and give some illustrative examples.
Keywords: Ky Fan inequalities, monotone, fixed point, nonexpansive mappings.
AMS Subject Classification: 2010, 65 K10, 90 C25.

1.

Introduction

Let C be a nonempty closed convex subset of Rn , the mapping T : C → C be
nonexpansive, i.e., ∥T (x) − T (y)∥ ≤ ∥x − y∥ for all x, y ∈ C, and let f : C × C → R
be a bifunction such that f (x, x) = 0 for all x ∈ C. We consider the Ky Fan
inequality over the fixed point set (see [9]), shortly KF (f, F ix(T )), as the follows:
Find x∗ ∈ F ix(T ) such that f (x∗ , y) ≥ 0 ∀y ∈ F ix(T ),
where F ix(T ) := {x ∈ C : T (x) = x}. The set of solutions of the problem is
denoted by Sol(f, F ix(T )). Problem KF (f, F ix(T )) is a special class of the Ky
Fan inequality on the nonempty closed convex constraint set. There are many
iterative methods for solving such problems which have been presented in [1, 5, 8,
10, 15, 16, 21, 23]. Popular applications of these problems are the power-control
problem for code-division multiple-access (shortly, CDMA) systems (see [12]) and
the Cournot-Nash oligopolistic market equilibrium model (see [6, 7, 13, 18]).
It is well-known that the gradient projection method in [27] solves the convex

This work was completed at the Vietnam Institute for Advanced Study in Mathematics and funded by
Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number
”101.02-2013.03”.
a Email:
ISSN: 0233-1934 print/ISSN 1029-4945 online
c 2014 Taylor & Francis
⃝
DOI: 10.1080/0233193YYxxxxxxxx



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optimization problem:
min{f (x) : x ∈ C},

(1)

where Ci is a closed convex subset of Rn for all i = 1, · · · , m, C := ∩m
i=1 Ci , and f is
a differentiable convex function on C. The iteration sequence {xk } of the method
is defined by xk+1 := P rC (xk − λ∇f (xk )). When C is arbitrary closed convex,
in general, computation of the metric projection P rC is not necessarily easy and
hence it is not effective for solving the convex optimization problem. To overcome
this drawback, Yamada in [25] proposed a fixed point iteration method
xk+1 := T (xk − λk ∇f (xk )),
∑
where T is a nonexpansive
mapping defined by T (x) := m
i=1 βi P rCi (x) for all x ∈
∑m
C, βi ∈ (0, 1) such that i=1 βi = 1. Under certain parameters βi (i = 1, · · · , m),
the sequence {xk } converges to a solution of Problem (1). Also this method has
applied for signal processing problems (see [11, 24]). Motivated by the fixed point
iteration method, Iiduka and Yamada in [12] proposed a subgradient-type method
for the equilibrium problems over the fixed point set of a nonexpansive mapping

EP (f, F ix(T )) and applied for the Nash equilibrium problem in noncooperative
games. The sequence {xk } is given by
 1
x ∈ Rn , ρ1 := ∥x1 ∥,



y k ∈ K := {x ∈ Rn : ∥x∥ ≤ ρ + 1}, f (y k , xk ) ≥ 0,
k
k
k ) ≤ f (y k , xk ) + ϵ ,

max
f
(y,
x
y∈Kk


(
) k
 k+1
x
:= T xk − λk f (y k , xk )ξ k , ρk+1 := max{ρk , ∥xk+1 ∥}, ξk+1 ∈ ∂f (y k , ·)(xk ),
where the sequences ∩
{ϵk }, {λ
and an asymtotic opk } were chosen appropriately,
}
∞ {
k

timization condition k=1 u ∈ F ix(T ) : f (y , u) ≤ 0 ̸= ∅ is satisfied. The authors showed that the iterative sequence {xk } converged to a point in Problem
KF (f, F ix(T )) without the metric projection onto a closed convex set.
The ergodic iteration technique is known to be a powerful tool for analyzing,
solving monotone variational inequalities (see [17, 19, 20, 25] and the references
quoted therein) and recently solving variational inequalities on the fixed point
set of nonexpansive mappings (see [11]). When T is the identity and f (x, y) =
⟨F (x), y − x⟩ where F : C → R, Problem KF (f, F ix(T )) is formulated as the
variational inequality, shortly V I(F, C), as the follows:
Find x∗ ∈ C such that ⟨F (x∗ ), x − x∗ ⟩ ≥ 0 ∀x ∈ C.
For solving V I(F, C), Ronald and Bruck [19] introduced an ergodic iteration
method which is very simple as the follows:


x1 ∈ C,




xk+1 = P rC (xk − λk F (xk )),



zk =




k
∑

λj x j

j=1
k
∑

.
λj

j=1

Under assumptions that the solution set of the problem is nonempty, F is monotone


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on C, and
{λk } ⊂ (0, ∞),

∞
∑

j=1

λj = ∞,

∞
∑

∥λj wj ∥2 < ∞,

j=1

the sequence {z k } converges to a solution of Problem V I(F, C). Note that, for the
monotone problem V I(F, C), the sequence {xk } may not be convergent.
In this paper, we propose new and simple algorithms for solving the Ky Fan
inequality over the fixed point set of nonexpansive mappings KF (f, F ix(T )). To
this problem, most of current algorithms are based on the strong monotonicity or
Lipschitz-type continuity of the bifunction f . The fundamental difference here is
that, at each main iteration in the proposed algorithms, we only require solving a
strongly convex auxilary problem and computing an ergodic iteration point with
only monotone assumption of f . Moreover, by choosing suitable regularization
parameters, we show that the iterative sequence globally converges to a solution of
Problem KF (f, F ix(T )).
The paper is organized as follows. Section 2 recalls some concepts related to
the Ky Fan inequality over the fixed point set of nonexpansive mappings, that
will be used in the sequel and new iteration algorithms. Section 3 investigates the
convergence theorem of the iteration sequences presented in Section 2 as the main
results of our paper.

2.

Preliminaries

We list some well known definitions of the bifunction and the projection under the
Euclidean norm, which will be required in our following analysis.
Definition 2.1: Let C be a nonempty closed convex subset of Rn , we denote
the metric projection on C by P rC (·), i.e,
P rC (x) = argmin{∥y − x∥ : y ∈ C} ∀x ∈ Rn .
The bifunction f : C × C → R ∪ {∞} is said to be
(I) monotone on C if for each x, y ∈ C,
f (x, y) + f (y, x) ≤ 0;
(II) Lipschitz-type continuous on C with constants c1 > 0 and c2 > 0 if for each
x, y ∈ C,
f (x, y) + f (y, z) ≥ f (x, z) − c1 ∥x − y∥2 − c2 ∥y − z∥2 .
By the definition, P rC satisfies the following property:
⟨x − P rC (x), P rC (x) − y⟩ ≥ 0 ∀x ∈ Rn , y ∈ C.
In this section, we assume that the bifunction f : C ×C → R and the nonexpansive
mapping T : C → C satisfy the following conditions:
(i) for each x ∈ C, f (x, ·) is continuous, convex and subdifferentiable on C;
(ii) f is monotone;
(iii) the solution set of Problem KF (f, F ix(T )) is nonempty.


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Algorithm 1.
Step 0. Choose positive sequences {λk } and {αk } such that 0 < λk+1 ≤ λk ,
∞
∑

λk = ∞, αk ≤ 1 and lim αk = 0. Take x1 ∈ C and k = 1.
k→∞

k=1

1
Step 1. Solve y k = argmin{λk f (xk , y) + ∥y − xk ∥2 : y ∈ C}.
2
Set xk+1 = αk xk + (1 − αk )T (y k ) and go to Step 2.
Step 2. Compute
k
∑
k
z(l)
=

λj xj

j=l
k
∑

(l = 1, · · · , k), k := k + 1 and come back to Step 1.
λj

j=l

Note that if ∥xk − y k ∥ = 0 then xk is a solution of the Ky Fan inequality KF (f, C)
(see [3, 4]). In this case, xk may not be a fixed point of T .
To analyse the convergence of Algorithm 1, we need to use the following technical
lemmas.
Lemma 2.2: (see [2]) Let {ak }, {bk } and {ck } be three sequences of nonnegative
real numbers satisfying the inequality:
ak+1 ≤ (1 + bk )ak + ck ,
∞
∑

for all integer k ≥ 1, where

bk < ∞ and

k=1

∞
∑

ck < ∞. Then, lim ak exists.
k→∞

k=1

Lemma 2.3: (see [20]) Let {ah } and {βh } be two sequences of nonnegative real

numbers satisfying the conditions:

lim ah = a ∈ R and

h→∞

∞
∑

βh = ∞.

h=1

Then, we have
h
∑

lim

h→∞

βj aj

j=1
h
∑

= a.
βj

j=1

3.

Convergent theorem

Now, we prove the main convergence theorems.


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Theorem 3.1 : Assume that assumptions (i) − (iii) hold, the sequence {xk } in
Algorithm 1 and {λk } satisfy
M :=

∞
∑

λk |f (xk , y k )| < ∞,

(2)

k=1

and there exists a positive number k0 such that
Sol(f, F ix(T )) ⊂ Ω := {x ∈ F ix(T ) : f (xk , x) ≤ 0 ∀k ≥ k0 }.

(3)

k . Claim that
Put z k := z(k
0)

(a) the sequences {xk } and {z k } are bounded;
(b) lim ∥xk+1 − T (xk )∥ = 0;
k→∞

(c) the sequence {z k } converges to a solution zˆ ∈ Sol(f, F ix(T )).
Remark 1 : If the bifunction f is H¨older continuous on C × C, i.e., there exist
constants Q > 0 and τ ∈ (0, 2] such that
|f (x, y) − f (x′ , y ′ )| ≤ Q∥(x, y) − (x′ , y ′ )∥τ ∀x, y, x′ , y ′ ∈ C.
Choosing {λk } such that
λk > 0,

∞
∑

λk = ∞,

k=1

∞
∑

2

λk2−τ < ∞.

k=1

Then, the condition (2) is satisfied. Indeed, since f is H¨older continuous on C × C,
we have
|f (xk , y k )| = |f (xk , y k ) − f (y k , y k )| ≤ Q∥(xk , y k ) − (y k , y k )∥τ = Q∥xk − y k ∥τ .
Combining this and (??) that
∥y k − xk ∥2 ≤ −λk f (xk , y k ) = λk |f (xk , y k )|,
we get
2

λk |f (xk , y k )| ≤ (Qλk ) 2−τ .
Consequently, the condition (2) is satisfied.
Proposition 3.2: Assume that assumptions (i) − (iii) hold, the sequence
{xk }, {y k } in Algorithm 1, {λk } satisfy the conditions (2) and there exists k0 ≥ 0
such that
∥xk − y k ∥ = o(λk ) ∀k ≥ k0 .
Then, the claims (a), (b) and (c) in Theorem 3.1 hold.
Proposition 3.3: Assume that assumptions (i) − (iii) hold, the sequence
{xk }, {y k } in Algorithm 1, {λk } satisfy the conditions (2) and there exists k0 ≥ 0
such that T is quasi nonexpansive with respect to W := {xk : k ≥ k0 }, i.e.,


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∥T (x) − xk ∥ ≤ ∥x − xk ∥ for all x ∈ C and xk ∈ W . Then, the claims (a), (b) and
(c) in Theorem 3.1 hold.
Using Algorithm 1 and Theorem 3.1, we obtain the following convergent theorem
for solving the Ky Fan inequality KF (f, C).
Corollary 3.4: Suppose that Assumptions (i) − (ii) are satisfied and Sol(f, C) ̸=
∅, x1 ∈ C and two positive sequences {λk }, {αk } satisfy the following restrictions:

∞
∑

λk+1 ≤ λk ,
λk = ∞,
k=1


αk ≤ 1, lim αk = 0.
k→∞

The sequences {xk } and {z k } are given by



xk+1 = argmin{λk f (xk , y) + 21 ∥y − xk ∥2 : y ∈ C},



k
∑

z k




=

λj x j

j=1
k
∑

.
λj

j=1

If the sequences {λk } and {xk } satisfy the conditions:
{λk } ⊂ (0, ∞),

∞

∑
k=1

λk = ∞,

∞
∑

λk |f (xk , xk+1 )| < ∞,

k=1

then {z k } converges to a point z ∗ ∈ Sol(f, C).
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