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

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

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 Anh a

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 Lipschitz-type 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 R n , 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: anhpn@ptit.edu.vn

ISSN: 0233-1934 print/ISSN 1029-4945 online c

⃝ 2014 Taylor & Francis

DOI: 10.1080/0233193YYxxxxxxxx

optimization problem:

where C iis a closed convex subset ofR n for all i = 1, · · · , m, C := ∩ m

i=1 Ci , and f is

a differentiable convex function on C The iteration sequence {x k } of the method

is defined by x k+1 := P r C (x k − λ∇f(x k )) When C is arbitrary closed convex,

in general, computation of the metric projection P r C 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

x k+1 := T (x k − λk∇f(x k )), where T is a nonexpansive mapping defined by T (x) :=∑m

i=1 βiP rC i (x) for all x ∈

C, β i ∈ (0, 1) such that ∑m

i=1 β i = 1 Under certain parameters β i (i = 1, · · · , m),

the sequence {x k } 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{x k } is given by











x1 ∈ R n , ρ1:=∥x1∥,

y k ∈ Kk:={x ∈ R n: ∥x∥ ≤ ρk+ 1}, f(y k , x k)≥ 0,

maxy ∈K k f (y, x k)≤ f(y k , x k ) + ϵ k ,

x k+1 := T(

x k − λk f (y k , x k )ξ k)

, ρ k+1:= max{ρk , ∥x k+1 ∥}, ξk+1 ∈ ∂f(y k , ·)(x k ),

where the sequences {ϵk}, {λk} were chosen appropriately, and an asymtotic

op-timization condition ∩∞

k=1

{

u ∈ F ix(T ) : f(y k , u) ≤ 0} ̸= ∅ is satisfied The

au-thors showed that the iterative sequence {x k } 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,

x k+1 = P r C (x k − λk F (x k )),

z k=

k

∑

j=1

λ j x j

k

∑

j=1

λ j

.

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

on C, and

{λk} ⊂ (0, ∞),

∞

∑

j=1

λj =∞,

∞

∑

j=1

∥λj w j ∥2< ∞,

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 {x k } 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

the metric projection on C by P r C(·), i.e,

P rC (x) = argmin {∥y − x∥ : y ∈ C} ∀x ∈ R n

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 r C satisfies the following property:

⟨x − P rC (x), P r C (x) − y⟩ ≥ 0 ∀x ∈ R n , 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.

Algorithm 1.

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

∞

∑

k=1

λ k=∞, αk ≤ 1 and lim

k →∞ α k = 0 Take x

1 ∈ C and k = 1.

Step 1 Solve y k = argmin {λk f (x k , y) +1

2∥y − x k ∥2 : y ∈ C}.

Set x k+1 = α kx k+ (1− αk )T (y k ) and go to Step 2.

Step 2 Compute

z (l) k =

k

∑

j=l

λj x j k

∑

j=l λj

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

Note that if∥x k −y k ∥ = 0 then x k is a solution of the Ky Fan inequality KF (f, C) (see [3, 4]) In this case, x k 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:

a k+1 ≤ (1 + bk )a k + c k ,

for all integer k ≥ 1, where ∑∞

k=1

bk < ∞ and ∑∞

k=1

ck < ∞ Then, lim

k →∞ ak exists.

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

lim

h →∞ a h = a ∈ R and ∑∞

h=1

β h =∞.

Then, we have

lim

h →∞

h

∑

j=1

β j a j h

∑

j=1

β j

= a.

3 Convergent theorem

Now, we prove the main convergence theorems

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

M :=

∞

∑

k=1 λk|f(x k , y k)| < ∞, (2)

and there exists a positive number k0 such that

Sol(f, F ix(T )) ⊂ Ω := {x ∈ F ix(T ) : f(x k , x) ≤ 0 ∀k ≥ k0}. (3)

Put z k := z k (k

0 ) Claim that (a) the sequences {x k } and {z k } are bounded;

(b) lim

k →∞ ∥x k+1 − T (x k)∥ = 0;

(c) the sequence {z k } converges to a solution ˆz ∈ 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

λ k > 0,

∞

∑

k=1

λ k=∞,∑∞

k=1 λ

2

−τ

k < ∞.

Then, the condition (2) is satisfied Indeed, since f is H¨ older continuous on C × C,

we have

|f(x k , y k)| = |f(x k , y k)− f(y k , y k)| ≤ Q∥(x k , y k)− (y k , y k)∥ τ = Q ∥x k − y k ∥ τ

Combining this and (??) that

∥y k − x k ∥2 ≤ −λk f (x k , y k ) = λ k|f(x k , y k)|,

we get

λ k|f(x k , y k)| ≤ (Qλk) −τ2 .

Consequently, the condition (2) is satisfied

{x k }, {y k } in Algorithm 1, {λk} satisfy the conditions (2) and there exists k0 ≥ 0 such that

∥x k − y k ∥ = o(λk) ∀k ≥ k0 Then, the claims (a), (b) and (c) in Theorem 3.1 hold.

{x k }, {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 := {x k : k ≥ k0}, i.e.,

∥T (x) − x k ∥ ≤ ∥x − x k ∥ for all x ∈ C and x k ∈ 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:







λ k+1 ≤ λk ,∑∞

k=1

λ k=∞,

α k ≤ 1, lim

k →∞ α k = 0.

The sequences {x k } and {z k } are given by











x k+1 = argmin {λk f (x k , y) +12∥y − x k ∥2: y ∈ C},

z k=

k

∑

j=1

λ j x j

k

∑

j=1

λ j

.

If the sequences {λk} and {x k } satisfy the conditions:

{λk} ⊂ (0, ∞),∑∞

k=1

λ k=∞,∑∞

k=1

λ k|f(x k , x k+1)| < ∞,

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

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