1 Introduction and preliminaries
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1 Proximal Methods for a Class of Relaxed Nonlinear Variational Inclusions Abdellatif Moudafi Université des Antilles et de la Guyane, Grimaag B.P. 7209, Schoelcher, Martinique abdellatif.moudafi@martinique.univ-ag.fr Abstract. Relying on the relaxed γ-r-cocoercivity notion introduced by Verma [8], we establish the convergence of relaxed proximal methods for a class of nonlinear variational inclusions. We obtain as a particular case the main result in [8] and some known results in this field. Furthermore, a weak convergence result for the proxgradient method is provided under a new condition which is weaker than cocovercivity. AMS subject classification: Primary, 49J53, 65K10; Secondary, 49M37, 90C25. Key words. Variational inequality, cocoercivity, hypomonotonicity, proximal method. 1 Introduction and preliminaries In a recent paper [8], Verma has proved the strong convergence of a modified projection-gradient algorithm under a relaxed cocoercive assumption, namely x,y H B(x) B(y),x y γ B(x) B(y) 2 +r x y 2. (1.1) The aim of this note is twofold. First, is to extend Verma s idea to more general problems. Specifically, given two operators A and B, we consider the problem of finding a zero of the sum of A and B. The splitting of an operator into the sum of two elementary operators A and B has usually a deep physical or economical meaning. The reason is that A and B may have very distinct properties. The second is more original, we first introduce a dual form of assumption (1.1), and then we prove a convergence result for the prox-gradient method. Tobeginwith, letusrecallthefollowingconceptswhichareofcommonuseinthe context of convex and nonlinear analysis. Throughout, H is a real Hilbert space,, denotes the associated scalar product and stands for the corresponding norm. An operator with domain D(A) is said to be monotone if u v,x y 0 whenever u A(x),v A(y). 1
2 2 A. Moudafi It is said to be maximal monotone if, in addition, the graph, gpha := {(x,y) H H : y A(x)}, is not properlycontainedin the graphofany othermonotone operator. It is well-known that for each x H and λ > 0 there is a unique z H such that x (I +λa)z. The single-valued operator Jλ A := (I +λa) 1 is called the resolvent of A of parameter λ. It is a nonexpansive mapping which is everywhere defined. Finally, recall that the inverse A 1 of A is the operator defined by x A 1 (y) y A(x). In what follows, we will focus our attention on the problem of finding a zero of the sum of two operators A and B on a real Hilbert space H, namely (P) find x H such that A(x)+B(x) 0, where A is a maximal monotone operator and B a nonlinear mapping on D(A). It is worth mentioning that when A = N C the normal cone to a convex set C, (P) amounts to finding a solution of a variational inequality, namely (VI) find x H; such that B(x ),x x 0 x K. It is well known that this problem has many applications in mechanics, economics, finance, network problems, management sciences and other branches of mathematical and engineering sciences. A fondamental approach to solving (P) when B is single-valued, is to consider the prox-gradient method, namely x k+1 = J A λ k (x k λ k B(x k )). In this note, based on Verma [8], we will also consider the following modified version x k+1 = (1 α k )x k +α k J A λ k (x k λ k B(x k )), (1.2) where λ k is a sequence of positive reals and 0 α k 1, k=0 α k =. 2 Convergence of proximal methods In this section we present the convergence of the proximal methods in the general context of the approximation of solution to the inclusion problem (P). 2.1 Strong convergence results Proposition 2.1 Let A be a maximal monotone operator and B : D(A) H be a relaxed γ-r-cocoercive nonlinear mapping which is also µ-lipschitz continuous with r > γµ 2. If the problem (P) has at least one solution, then the sequence (x k ) generated by (1.2) strongly converges to an element x which solves (P) provided that the sequence (λ k ) satisfies the condition 0 < λ k < 2(r γµ 2 )/µ 2.
3 Proximal method for a class of relaxed variational inclusions 3 Proof. Since x is a solution of (P), it follows that x = J A λ k (x λ k B(x )). From (1.2) and according to the convexity of the norm, we have x k+1 x = (1 α k )(x k x ) + α k (J A λ k (x k λ k B(x k )) J A λ k (x λ k B(x )) (1 α k ) x k x + α k J A λ k (x k λ k B(x k )) J A λ k (x λ k B(x )) (1 α k ) x k x +α k x k x λ k (B(x k ) B(x )). Using the fact that B is relaxed γ-r-cocoercive and µ-lipschitz continuous, we have successively x k x λ k (B(x k ) B(x )) 2 = x k x 2 2λ k B(x k ) B(x ),x k x + λ 2 k B(x k) B(x ) 2 Finally, we obtain x k x 2 +2λ k γ B(x k ) B(x ) 2 2 rλ k x k x 2 +λ 2 k µ2 x k x 2 (1 2λ k r+2λ k γµ 2 +λ 2 k µ2 ) x k x 2. x k+1 x (1 α k ) x k x +α k θ x k x, where θ = 1 2λ k r+2λ k γµ 2 +λ 2 k µ2. From which deduce which in turn ensures x k+1 x (1 (1 θ)α k ) x k x, x k+1 x Π k i=1 (1 (1 θ)α i) x 0 x. Since θ < 1 and k=0 α k =, it implies thanks to [9], that lim k + Πk i=1(1 (1 θ)α i ) = 0. Hence, (x k ) strongly converges to x for 0 < λ k < 2(r γµ 2 )/µ 2. This completes the proof. We would like to emphasize that in the particular case where A = f, f being a proper convex lower semicontinuous function, (P) is nothing but the variational inequality: find x H such that (GVI) B(x ),x x +f(x) f(x ) 0 x H,
4 4 A. Moudafi and algorithm (1.2) amounts to x k+1 = (1 α k )x k +α k prox λk f(x k λ k B(x k )), where prox λf (x) = argmin y H {f(y) + 1/2λ y x 2 }. In this context, we recover well-known results (see for example [2]) by setting γ = 0. Moreover, if f = δ C the indicator function of a closed convex set C and γ > 0 (resp. γ = 1), (P) reduces to (IV) and the algorithm to x k+1 = (1 α k )x k +α k P C (x k λ k B(x k )), and we recover the main result of Verma [8] (resp. [7]). 2.2 A weak convergence result For simplicity sake, we take in this section α k = 0 k IN. More precisely, we consider the following prox-gradient algorithm x k+1 = J A λ k (x k λ k B(x k )), (2.3) where λ k is a sequence of positive reals having to tend to zero. The convergence of this method will be studied under a relaxed γ-r-coercivity condition, namely B(x) B(y),x y γ B(x) B(y) 2 r x y 2, (2.4) which is clearly weaker than the cocoercivity. Furthermore, it is easy to check that relaxed γ-r-coercivity implies Lipschitz continuity. Indeed, from relation (2.4) and by using Cauchy-Schwarz inequality, we have B(x) B(y) 2 γ 1 B(x) By) x y +γ 1 r x y 2, which, by an elementary computation, implies that B(x) B(y) γr x y. 2γ Note that in addition to the fact that B is Lipschitz continuous, B is a fortiori r-hypomonotone and so is A+B, thus the graph of A+B is weakly-strongly closed and the solutionset of(p) is closed and convexsee for example [1]. These properties will be needed for proving the convergence of the whole sequence. Proposition 2.2 Let A be a maximal monotone operator and B : D(A) H be a relaxed γ-r-coercive nonlinear mapping. Assume that problem (P) has at least one solution and suppose that the sequence (x k ) generated by (2.3) is asymptotically regular in the sense that lim k λ 1 k x k+1 x k = 0. Then (x k ) weakly converges to an element x which solves problem (P) provided that + k=0 λ k < + and 0 < λ k 2γ.
5 Proximal method for a class of relaxed variational inclusions 5 Proof. Since x is a solution of (P), it follows that x = J λk (x λ k B(x )). From (2.3), we have x k+1 x = J A λ k (x k λ k B(x k )) J A λ k (x λ k B(x )) x k x λ k (B(x k ) B(x )). According to the fact that B is relaxed γ-r-coercive, we have x k+1 x 2 = x k x 2 2λ k B(x k ) B(x ),x k x + λ 2 k B(x k ) B(x 2 Since 0 < λ k 2γ and x k x 2 2λ k γ B(x k ) B(x ) 2 +2 rλ k x k x 2 +λ 2 k B(x k ) B(x ) 2 (1+2λ k r) x k x 2 λ k (2γ λ k ) B(x k ) B(x ) 2. λ k < +, in the light of lemma 1 in [6], we have k=0 that lim k + x k x 2 exists and consequently the sequence (x k ) is bounded. Let x be weak cluster point of (x k ) and (x ν ) a subsequence weakly converging to x. Since λ k 2γ and B is Lipschitz continuous, the asymptotical regularity condition of the sequence (x k ) ensures that lim B(x k) B(x k+1 ) = 0. k + Now algorithm (2.3) can be rewritten as λ 1 k (x k x k+1 ) (B(x k ) B(x k+1 )) (A+B)(x k+1 ). Passing to the limit in the last inclusion and taking into account the fact that the graph of the operator A+B is weakly-stronglyclosed, we obtain at the limit 0 (A+B)( x), in other words, every weak-cluster point of the sequence (x k ) is a solution of (P). The convergence of the whole sequence follows by applying opial s lemma (see [3]). This completes the proof. Remark 2.1 It is worth mentioning that in this case the main operator which to x associates J A λ k (x λ k B(x)) is asymptotically nonexpansive and thus we can obtain other convergence results by using fixed-point theory, (see for example [5]).
6 6 A. Moudafi References [1] A.N. Iusem, T. Pennanen and B. F. Svaiter, Inexact variants of the proximal point algorithm without monotonicity. SIAM Journal on optimization, 13 (4), (2003), p [2] A. Moudafi and M. Théra, Finding the zero for the sum of two maximal monotone operators, J. Optim. Theory & Appl., vol. 94, N2, (1997), [3] Z. Opial, Weak convergence of successive approximations for nonexpansive mappings. Bull. of the American Math. Soc. 73, (1967), p [4] R. T. Rockafellar and R. Wets, Variational Analysis, Springer, Berlin [5] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings.. J. Math. Anal. Appl. 158, No.2 (1991), p [6] K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by Ishikawa iteration process. J. Math. Anal. Appl. 178 (1993), p [7] R.-U. Verma, Projection methods, algorithms and new systems of variational inequalities, Computers and Mathematics with Applications 41 (2001) p [8] R.-U. Verma, A class of relaxed γ-r-cocoercive nonlinear variational inequalities and convergence of projection methods, Mathematical Inequalities and Applications 7 (2) (2004) p [9] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. der Mathematik 58 (1992), p
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