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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
December 23, 2014 17:44 Optimization KyFanonFixOptim˙Copy 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: anhpn@ptit.edu.vn ISSN: 0233-1934 print/ISSN 1029-4945 online c 2014 Taylor & Francis ⃝ DOI: 10.1080/0233193YYxxxxxxxx http://www.informaworld.com December 23, 2014 17:44 Optimization KyFanonFixOptim˙Copy Pham N. Anh and Trinh N. Hai 2 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 December 23, 2014 17:44 Optimization KyFanonFixOptim˙Copy Optimization 3 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. December 23, 2014 17:44 Optimization KyFanonFixOptim˙Copy Pham N. Anh and Trinh N. Hai 4 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. December 23, 2014 17:44 Optimization KyFanonFixOptim˙Copy Optimization 5 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., December 23, 2014 17:44 Optimization KyFanonFixOptim˙Copy REFERENCES 6 ∥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). References [1] P. N. Anh, A new extragradient iteration algorithm for bilevel variational inequalities, Acta Mathematica Vietnamica, 37 (2012), 95-107. [2] P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optim., 62 (2013), 271-283. [3] P. N. Anh, Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities, J. Optim. Theory Appl., 154 (2012), 303 - 320. [4] P. N. Anh, and J. K. 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Prog., 11 (1976), 128-149. ... 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... that the iterative sequence globally converges to a solution of Problem KF (f, F ix(T )) The paper is organized as follows Section recalls some concepts related to the Ky Fan inequality over the. .. 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