ENLARGEMENT OF MONOTONE OPERATORS WITH APPLICATIONS TO VARIATIONAL INEQUALITIES. Abstract

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1 ENLARGEMENT OF MONOTONE OPERATORS WITH APPLICATIONS TO VARIATIONAL INEQUALITIES Regina S. Burachik* Departamento de Matemática Pontíficia Universidade Católica de Rio de Janeiro Rua Marques de São Vicente, 225 Rio de Janeiro, RJ, CEP ,Brazil address: Alfredo N. Iusem** B. F. Svaiter*** address: Instituto de Matemática Pura e Aplicada Estrada Dona Castorina, 110 Rio de Janeiro, RJ, CEP , Brazil Abstract Given a point-to-set operator T, we introduce the operator T ε defined as T ε (x) = {u : u v, x y ε for all y R n, v T (y)}. When T is maximal monotone T ε inherits most properties of the ε-subdifferential, e.g. it is bounded on bounded sets, T ε (x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of T ε, and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consists of solving problems of the form 0 H ε (x), while the subproblems of the exact method are of the form 0 H(x). If ε k is the coefficient used in the k-th iteration and the ε k s are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is enough well behaved, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved. Keywords: Convex optimization, variational inequalities, proximal point methods, monotone operators AMS Classification numbers: 90C25, 90C30. *Research of this author was supported by CNPq grant no /94-0 **Research of this author was partially supported by CNPq grant no /86 ***Research of this author was supported by CNPq grant no /93-9(RN)

2 1. Introduction Given a convex function f: R n R { }, the subdifferential of f at x, i.e. the set of subgradients of f at x, denoted by f(x), is defined as f(x) = {u R n : f(y) f(x) u, y x 0 for all y R n }. An extension of this concept is the notion of ε-subdifferential, denoted as ε f(x) and defined, for ε 0, as ε f(x) = {u R n : f(y) f(x) u, y x ε for all y R n }. This notion has proved to be quite useful in connection with optimization algorithms (see [8], [13]): in many cases in which an algorithm requires computation of an element of f(x), or of a zero of f (i.e. an x such that 0 f(x), which is a minimizer of f) the convergence properties of the method are preserved when f is replaced by ε f for some adequate ε > 0. Since in general f(x) ε f(x), such replacement gives at the same time more latitude and more robustness to the method, taking into account that exact computation is not achievable in computer implementations. In this paper, we introduce a similar enlargement of a monotone operator T, which we denote by T ε and define as T ε (x) = {u R n : u v, x y ε for all y R n, v T (y)}. When T = f for some f, we do not get ε f = T ε but rather ε f T ε. In Section 2 we establish several properties of T ε, like the one just mentioned. Most properties valid for ε f extend to T ε under suitable assumptions on T. For instance, when T is uniformly monotone for all ε > 0 there exist radiuses δ, δ such that T ε (x) is in between the images through T of the balls with centers at x and such radiuses. We want to point out previous works in enlargements of monotone operators for solving the variational inequality problem. For instance, a different regularization process with monotone perturbations of the original maximal monotone operator has been considered in [1]. In [18] and [19], Liskovets studied a nonmonotone perturbation based on the notion of Hausdorff distance. Next, we proceed to develop an application of the new concept in relation with proximal point methods for variational inequalities. Given a convex set C and a point-to-set monotone operator T, the variational inequality problem (VIP(T, C) from now on), consists of finding z C such that there exists u T (z) satisfying u, x z 0 for all x C. VIP(T, C) reduces to minimizing f(x) subject to x C when T (x) = f(x) with f a convex function, and to solving the equation 0 T (x) when C = R n. The methods of interest, introduced in [4], generate a sequence {x k }, where x k+1 is the unique zero of an operator of the form T + H k, with H k (x) = λ k ( g(x) g(x k )), for some λ k > 0 and some strictly convex function g whose gradient diverges at the boundary of the constraint set C. The function g is said to be a Bregman function with zone C 0 (the interior of C). In Section 3 we introduce some material required for the analysis of 2

3 these methods. These algorithms extend the standard proximal point method for finding zeroes of monotone operators, corresponding to C = R n and g(x) = x 2 (see [22]), to the constrained case, i.e. to VIP(T, C), and transform the original constrained problem into a sequence of unconstrained (or perhaps more easily constrained) subproblems. In Section 4 we use the concept introduced in Section 2 to present a relaxation of the methods studied in [4]. Now the subproblems do not consist any more of finding exactly the unique zero of T + H k, but rather any zero of T ε k + H k, with k=0 ε k <. In this sense, the relaxed method can be seen as an inexact version of the method in [4]. We will prove that the convergence properties of the relaxed method are the same as those of the exact one, showing that it is rather robust with respect to numerical errors. Under adequate regularity assumptions on T (e.g. uniform monotonicity) we prove that the set of solutions of the relaxed subproblem contains a ball with center in the exact solution of the unrelaxed subproblem. This means that if the subproblem is solved with any method guaranteed to converge to a zero of a monotone operator (e.g. [9], [15]), a solution of a relaxed subproblem will be found after finitely many iterations. However, it is not clear at present how to exploit this fact in practice. Similar convergence results for an inexact version of the classical proximal point method are presented in [22]. In this case the radius of the ball is just ε k and no hypotheses on T are required beyond maximal monotonicity and existence of zeroes. This is a consequence of some particular properties of g(x) = x 2 (like the triangular inequality) which do not hold for other Bregman functions appropriate for the constrained case (C R n ). In the context of constrained problems, a related inexact proximal point method has been considered in [14] for the optimization case, i.e. with T = f, so that VIP(T, C) becomes min f(x) s.t. x C. Its iteration is of the form 0 ( εk f + H k )(x k+1 ). Since, as mentioned before, we have in general T ε ε f for T = f, our method, when applied to optimization problems, does not reduce to Kiwiel s. In fact, the sets of solutions of our subproblems are in general larger than the corresponding ones in Kiwiel s method. This is the case, for instance, of T as in Examples 2 and 3 in Section 2. Throughout the paper, superindex t denotes transpose,, and are the Euclidean inner product and norm respectively, g indicates the gradient of a differentiable function g and R n +, R n ++ are defined as R n + = {x R n : x j 0 for all j}, R n ++ = {x R n : x j > 0 for all j}. 2. Definition and properties of T ε. Recall that an operator T : R n P(R n ) is monotone if and only if u v, x y 0 for all x, y R n, and all u T (x), v T (y). T is maximal monotone if and only if it is monotone and, additionally, T (x) T (x) for all x R n and some monotone T implies T = T. Given T : R n P(R n ) and ε 0 define T ε : R n P(R n ) as T ε (x) = {u R n : u v, x y ε for all y R n, v T (y)}. (1) We start with some elementary properties of T ε which require no assumptions on T. The notation T T means T (x) T (x) for all x R n, and T +T is the operator defined as (T + T )(x) = {u + v : u T (x), v T (x)}. 3

4 Proposition 1. i) If ε 1 ε 2 0 then T ε 2 T ε 1. ii) T ε T ε 2 2 (T 1 + T 2 ) ε 1+ε 2. iii) T ε (x) is closed and convex for all x R n. iv) If 0 ε = lim k εk, x = lim k xk, u = lim k uk and u k T (x k ) for all k then u T ε (x). v) αt ε = (αt ) αε for all ε 0 and all α 0. vi) αt1 ε + (1 α)t2 ε (αt 1 + (1 α)t 2 ) ε for all ε 0, α [0, 1]. vii) If E R + then T ε = T ε with ε = inf E. Proof: Elementary. ε E The properties above hold for any T, but little else can be said about T ε without monotonicity of T, so from now on we will work with maximal monotone operators. Recall that dom T = {x R n : T (x) }, Im T = x R n T (x). Proposition 2. Assume that T is maximal monotone. Then i) T 0 = T, ii) If dom T is closed then dom T ε = dom T for all ε 0, iii) If dom T is closed them Im T ε Im T for all ε 0, iv) If dom T and Im T are both closed then Im T ε = Im T for all ε 0. Proof: i) Follows easily from the maximality property of T. ii) dom T dom T ε for all ε 0 by (i) and Proposition 1(i). dom T is closed by hypothesis and consequently convex (see [23, p. 915] or [2, Theorem 2.2]). Take x dom T ε and let y dom T be the projection of x onto dom T. By the projection property of y, x y, z y 0 for all z dom T. Take v T (y), z dom T, v T (z) and λ 0. Then v + λ(x y) v, y z = λ x y, y z + v v, y z 0 using a well known property of the projection and monotonicity of T. By maximality of T, v + λ(x y) T (y) for all λ 0. Take u T ε (x). Then ε u v λ(x y), x y = u v, x y λ x y 2 implying λ x y 2 u v, x y +ε for all λ 0, which is impossible unless x y 2 = 0, i.e. x = y. Since y dom T, we get x dom T, proving that dom T ε dom T. iii) Let P = Im T. P is convex (see [18, p. 915] or [2, Proposition 2.5]). Take u Im T ε. Then u T ε (x) for some x dom T ε. Let ū be the projection of u onto P. Take any λ 0 and define z = x + λ(u ū) (2) Let z be the projection of z onto dom T, and v the projection of u onto T ( z), which is closed and convex. By a well known property of the projection, u ū, v ū 0 for all v P, (3) 4

5 Since u T ε (x), v T ( z), z z, y z 0 for all y dom T, (4) u v, v v 0 for all v T ( z). (5) ε u v, x z = u v, z z λ u v, u ū = u v, z z λ u ū 2 λ ū v, u ū u v, z z λ u ū 2 (6) using (2) in the first equality and (3) in the inequality, and noting that v T ( z) P. Now take any y dom T, v T (y) and get v + z z v, z y = v v, z y + z z, z y 0 using monotonicity of T and (4). By maximality of T, v + z z T ( z), i.e. v + z z = ˆv for some ˆv T ( z). Then, using (5), u v, z z = u v, ˆv v 0. (7) From (6) and (7), ε λ u ū 2 implying 0 u ū 2 ε/λ. Since λ is an arbitrary positive real number, we get u = ū. Since ū P, we get u P. It follows that Im T ε Im T. iv) By (i) and Proposition 1(i), Im T Im T ε for all ε 0, and the result follows from (iii). We mention that in general neither T ε nor ε f are monotone operators. compare T ε with ε f in the case of T = f with f a convex function. Next we Proposition 3. If T = f with f a convex function then ε f(x) T ε (x) for all x R n. Proof: Take u ε f(x). Consider any y R n, v T (y) = f(y). By definition of ε f and f, Using (1), it follows that u T ε (x). ε + u, x y f(x) f(y) v, x y. One could expect to have in this case T ε = ε f or at least T ε ε f for some ε. This is not the case, as is shown in the following examples of explicit computation of T ε and ε f, which require just elementary calculus and some care. Example 1: n = 1, f(x) = p 1 x p with p 1, T = f. i) For p = 1 we have [1 ε x, 1] if x > ε 2 ε f(x) = T ε (x) = [ 1, 1] if x ε 2 [ 1, 1 ε x ] if x < ε 2 5

6 and ε f = T ε ii) For p = 2, T ε (x) = [x 2 ε, x + 2 ε], ε f(x) = [x 2ε, x + 2ε] so that T ε (x) = 2ε f(x). iii) For any p > 1 and x = 0, T ε ε ε (0) = [ p( p 1 )1 1/p, p( p 1 )1 1/p ], [ ε f(0) = ( pε p 1 )1 1/p, ( pε ] p 1 )1 1/p, so that T ε (0) = ε f(0) with ε = ε p 1/(p 1). Example 2: n = 1, f(x) = log x, T = f. For x > 0 [ 1 T ε (x) = x ( 1 ε 2 ε), 1 ] min{ 1 + ε, 0}, ε f(x) = x [ 1 x s 1(ε), 1 ] x s 2(ε) where 0 < s 2 (ε) 1 s 1 (ε) and s 1 (ε), s 2 (ε) are the two roots of s 1 log s = ε. More generally, for f(x) = α n j=1 log x j and T = f, with α > 0, it is easy to check that 0 T ε (x) for all x > 0 and all ε αn, while 0 / ε f(x) for any x > 0 and any ε > 0. Therefore, if ε αn then T ε (x) ε f(x) for any x > 0 and any ε 0. Example 3: f(x) = x t Qx with Q R n n symmetric and positive definite, T = f. Then ε f(x) = {Qx + w : w t Q 1 w 2ε}, T ε (x) = {Qx + w : w t Q 1 w 4ε}, so that T ε (x) = 2ε (x). Proposition 3 and the examples above can be seen as a favorable feature of T ε as compared to ε f: since T ε (x) is larger than ε f(x) for T = f, to find a vector in T ε (x) or to solve u T ε (x) for a fixed u is in principle easier than with ε f instead of T ε. But then we have to check that T ε does not enlarge T too much. The next propositions show that T ε is indeed not too large. For Q R n we will denote by T (Q) the set T (x) and by B(x, δ) the ball with center at x and radius δ. An operator T will be said to be locally bounded if for all x (dom T ) 0 (the interior of dom T ) there exists δ x such that T (B(x, δ x )) is bounded, and bounded on bounded sets if for all bounded set Q R n whose closure Q is contained in (dom T ) 0 it holds that T (Q) is bounded. We consider only points and sets in (dom T ) 0 because no maximal monotone operator can be bounded at a point in the boundary of its domain, where T (x) contains a nontrivial cone. An elementary compactness argument shows that any locally bounded operator is bounded on bounded sets: the balls B(x, δ x ) cover Q and if M x is an upper bound of the norm of any element of B(x, δ x ) then the maximum of M xi corresponding to a finite subcovering is an upper bound of u for any u T (Q). Proposition 4. If T is maximal monotone then T is locally bounded, and henceforth bounded on bounded sets. Proof: See [21, Theorem 1]. 6 x Q

7 For Q R n let M T (Q) = sup{ u : u T (Q)}. Proposition 5. If T is maximal monotone and dom T is closed then T ε is bounded on bounded sets for all ε 0. Proof: By Proposition 2(ii), (dom T ε ) 0 = (dom T ) 0. Take Q bounded and such that Q (dom T ) 0. Let d(x, Q) be the Euclidean distance from a point x to the set Q, and, for ν 0, let Q ν = {x R n : d(x, Q) ν}. Clearly there exists ν > 0 such that Q ν (dom T ) 0. We will prove that M T ε(q) ε ν +M T (Q ν )+2M T (Q) for any such ν. Note that M T (Q), M T (Q ν ) are finite by Proposition 4, so that the inequality above establishes ν u û the result. Take x Q, u T ε (x), û T (x). Write u = û + λ ν w with w = (u û), λ = u û (if u = û, take w = 0, λ = 0). Let y = x + w, so that x y = w = ν. Then, since u T ε (x), for all v T (y) ε u v, x y = λ ν w, x y + û v, x y = λ ν w 2 + û v, x y λν + û v x y λν + ( û + v )ν. (8) Since x y = ν and x Q, we have y Q ν. So û M T (Q), v M T (Q ν ). It follows from (8) that λν ε + νm T (Q) + νm T (Q ν ) and therefore u u û + û = λ + û ε ν + M T (Q ν ) + 2M T (Q). (9) Since u is an arbitrary element of T ε (Q), it follows from (9) that M T ε(q) ε ν + M T (Q ν ) + 2M T (Q) <. Next we proceed to look more closely at the relation between T ε (x) and T (B(x, δ)), which is basic for some applications of the new concept. First we need a simple continuity property of maximal monotone operators. Proposition 6. Let T be maximal monotone and take ˆx (dom T ) 0. For all θ > 0 there exists δ > 0 (depending on ˆx) such that for all x B(ˆx, δ) and all u T (x) there exists û T (ˆx) satisfying u û θ. Proof: If the result does not hold then there exists a sequence {x k } converging to ˆx, and u k T (x k ) such that u k û θ for all û T (ˆx). {u k } is bounded by Proposition 4. Refine the sequence {u k } so that it converges to some u, and get u û θ for all û T (ˆx). On the other hand, since the graph of T is closed (see [2]), use Proposition 1(iv) with ε k = ε = 0 and Proposition 2(i) to conclude that u T (ˆx), which is a contradiction. Corollary 2. If T is maximal monotone and point-to-point in (dom T ) 0 then T is continuous in (dom T ) 0. Proof: In the point-to-point case (i.e. u = T (x), û = T (ˆx), etc), Proposition 6 reduces to establishing continuity of T at ˆx. 7

8 We will also impose stronger hypotheses on T. A maximal monotone point-to-set operator T is said to be uniformly monotone if and only if there exist γ > 0, p > 1 such that u v, x y γ x y p for all x, y dom(t ), all u T (x) and all v T (y). Recall that any point-to-set operator T can be inverted: T 1 (x) is just the set {y R n : T (y) = x}. Uniformly monotone operators have the following fundamental property. Proposition 7. If T : R n P(R n ) is uniformly monotone then dom T 1 = R n and T 1 is point-to-point and Hölder-continuous. Proof: [23, Theorem 26.A]. Now we establish the main property of T ε. Theorem 1. Assume that T is maximal monotone and uniformly monotone. Take x (dom T ) 0. Then i) For all ε > 0 there exists δ > 0 such that T (B(x, δ)) T ε (x). ii) For all δ > 0 there exists ε > 0 such that T ε (x) T (B(x, δ)). Proof: i) Let [ ] 1 1/p ε β = pγ (10) γ(p 1) where p, γ are the constants of uniform monotonicity. Apply Proposition 6 with θ = β and ˆx = x. We will verify that the δ whose existence is ensured by Proposition 6 is enough for the result. Take x B(x, δ), u T (x). By Proposition 6 there exists u T (x) such that u u β. Take any y dom T and v T (y). Then u v, x y = u u, x y + u v, x y u u x y + γ x y p β x y + γ x y p (11) using uniform monotonicity in the first inequality. Let ϕ: R n R be defined as ϕ(t) = βt+γt p with β as in (10). Use elementary calculus to find the minimum of ϕ and conclude that ϕ(t) ε for all t 0. It follows from (11) that u v, x y ε for all y dom T and all v T (y), so that u T ε (x). We have proved that T (B(x, δ)) T ε (x). ii) First we claim that there exists σ > 0 such that d(u, T (x)) σ implies u T (B(x, δ)), where d, as before, denotes the Euclidean distance from a point to a set. Otherwise there exist sequences {u k } T (x), {u k } such that lim k (uk u k ) = 0 and u k / T (B(x, δ)), or equivalently T 1 (u k ) x δ, for all k. Since T (x) is bounded by Proposition 4, we may assume without loss of generality that u k converges to some u, and therefore u = lim k uk. By Propositions 1(iv) and 2(i) u belongs to T (x). By Proposition 7, T 1 (u k ) converges to T 1 (u) = x so that lim T 1 (u k ) x = 0. The resulting contradiction shows that the k claim holds. By Proposition 6 there exists η > 0 such that for all y B(x, η) and all v T (y) there exists û T (x) satisfying v û σ 2. Take ε ησ 2. We claim that T ε (x) T (B(x, δ)). Take u T ε (x). Let u be the orthogonal projection of u onto T (x), which is closed and 8

9 convex by Proposition 1(iii) and 2(i). If u = u then u T (x) T (B(x, δ)). Otherwise, write u = u + u u η w for some w R n. Let y = x + w and take v T (y). Note that w = η, so that y B(x, η). By definition of η, there exists û T (x) such that v û σ 2. Since u T ε (x) ε u v, x y = u u, x y + u û, x y + û v, x y = u u, w u û, w + û v, x y η = η u u + u û, u u + û v, x y. (12) u u Since u is the orthogonal projection of u onto T (x) and û T (x), the second term in the rightmost expression of (12) is nonpositive. Also, the third one is bounded above by û v x y ση ση 2. We conclude that ε η u u + 2, so that u u ε η + σ 2 σ 2 + σ 2 = σ. We have shown that d(u, T (x)) σ and therefore u T (B(x, δ)). Since u is any element of T ε (x) we have proved that for ε ησ 2 it holds that T ε (x) T (B(x, δ)). The uniform monotonicity hypothesis in Theorem [ 1 cannot ] be just dropped, as the 0 1 following examples show: if T (x) = Ax with A = then T 1 0 ε = T for all ε 0, so that T ε (0) = 0, while T (B(0, δ)) = B(0, δ) for all δ 0, showing that item (i) fails for this T ; if T is as in Example 1(i) then T (B(1, δ)) = {1} for all δ < 1, while max{0, 1 ε} T ε (1) for all ε > 0, showing that item (ii) fails for this T. In both cases, T is maximal monotone. In the sequel we will need a stronger version of Theorem 1(i), where the same δ does the job for all x in some neighborhood of x. We remark that Proposition 6 does not hold in this locally uniform way: if T (x) = f(x) with f(x) = x, we must take δ x x, because otherwise we can always find x B(ˆx, δ x ) with sg(x) = sg(ˆx), in which case u û = 2 for all u T (x), û T (ˆx). We will use a different line of proof, but then we need to impose some further condition besides uniform monotonicity: either dom T = R n or T is point-topoint. On the other hand, we will establish this stronger version for some operators which need not be uniformly monotone: those of the form T = f. Theorem 2. Assume that T is maximal monotone. Fix x (dom T ) 0. If either i) T = f for some convex function f, or ii) T is uniformly monotone and dom T = R n, or iii) T is uniformly monotone and point-to-point in (dom T ) 0, then there exists δ > 0 such that T (B(x, δ)) T ε (x) for all x B(x, δ). Proof: Let M t = sup{ u : u T (B(x, t))}. We will treat separately the three hypotheses on T. i) Take ν such that B(x, ν) (dom T ) 0. Define δ = 1 2 min {ν, ε M ν }. Note that M ν is finite by Proposition 4. Take x B(x, δ), x B(x, δ), u T (x ), u T (x). Observe that δ ν 2 implies x, x B(x, ν) (dom T ) 0. Then, for all y R n f(y) f(x) u, y x = f(y) f(x ) u, y x u, x x + f(x ) f(x) u, x x + u, x x ( u + u ) x x 2M ν x x 2M ν δ ε 9

10 using the subgradient property in the first inequality. We have proved that u ε f(x), i.e. that T (B(x, δ)) ε f(x). The result follows then from Proposition 3. ii) Now dom T = R n and we can start with an arbitrary ball around x, say of radius 1. ( Let ρ = 1 + 2M 1 γ ) 1/(p 1), where γ, p are the constants of uniform monotonicity, and take δ = 1 2 min {1, ε M ρ }. Note that M ρ is finite by Proposition 4. Take x B(x, δ), x B(x, δ), u T (x ), u T (x). We need to prove that u v, x y ε (13) for all y R n, v T (y). We consider two cases ( a) y x ρ. In this case y x y x x x ρ δ ρ 1 = so that Then u v, x y = u u, x y + u v, x y ) 1/(p 1) 2M 1 γ γ y x p 1 2M 1. (14) u u x y + γ x y p ( u + u ) x y + γ x y p (15) using uniform monotonicity of T in the last inequality. Since δ 1 2, we have x x 1, i.e. x, x B(x, 1) so that u M 1, u M 1. From (14) and (15) u v, x y 2M 1 x y + γ x y p = x y ( 2M 1 + x y p 1 ) 0 ε. We have proved that (13) holds. b) y x ρ. In this case v T (B(x, ρ)), so that v M ρ. M ρ is finite by Proposition 4, because dom T = R n. In this case u v, x y = u v, x x + u v, x y u v, x x ( u + v ) x x ( u + v )δ (16) using monotonicity of T in the first inequality. Note that x B(x, 1) B(x, ρ) because ρ > 1. It follows that u M ρ. We conclude from (16) that u v, x y 2M ρ δ ε and (13) holds also in this case. We remark that we need dom T = R n because we must have B(x, ρ) (dom T ) 0 in order to ensure finiteness of M ρ, and we cannot make ρ arbitrarily small, even if we start with a ball of radius t instead of 1, i.e. with M t instead of M 1 in the formula of ρ, because even M 0 can be strictly positive. iii) Now (13) becomes T (x ) T (y), x y ε. (17) Take ν such that B(x, ν) (dom T ) 0. By Corollary 2, T is continuous in B(x, ν) and therefore uniformly continuous. Take η such that for all x, x B(x, η) satisfying x x η it holds that T (x) T (x ) β with β as in (10). Take δ = 1 2 min{η, ν}. Since 10

11 x B(x, δ), x B(x, δ) it follows that x x η, so that T (x) T (x ) β. Then, as in the proof of Theorem 1(i), T (x ) T (y), x y = T (x ) T (x), x y + T (x) T (y), x y T (x ) T (x) x y + γ x y p β x y + γ x y p = ϕ( x y ) ε using uniform monotonicity of T in the first inequality, and therefore (17) holds. Corollary 3. If both (ii) and (iii) of Theorem 2 hold and furthermore T is Lipschitz continuous with constant L then T (B(x, δ)) T ε (x) with the explicit value of δ given by [ 1 1/p. δ = pγ ε L γ(p 1)] Proof: If x x δ then T (x ) T (x) L x x Lδ = β, with β as in (10), and then we continue as in the proof of item (iii) of Theorem 2. Next we look at our extension of T applied to approximate solutions of variational inequality problems, defined in Section 1. For a closed and convex set Q, define I Q as I Q (x) = 0 if x Q, I Q (x) = + otherwise, and let N C = I C. N C is called the normal cone operator of C. It is easy to check that the set of solutions of VIP(T, C) is the set of zeroes of T + N C and, more generally, that the solution set of VIP(T, C V ) coincides with the solution set of VIP(T + N V, C). In Section 4 we will consider an algorithm for solving VIP(T, D) with the following structure: write D = C V in such a way that C has nonempty interior and proceed to solve VIP(T + N V, C) by solving a sequence of subproblems of the type VIP(T + N V + H k, R n ), equivalent to VIP(T + H k, V ), where H k is a regularization operator which guarantees that the solutions of VIP(T + H k, V ) belong to the interior of C, so that they solve in fact VIP(T + H k, V C) = VIP(T + H k, D). The idea is to put in V those constraints which make the interior of D empty, if any (typically linear equalities), and to leave in C the remaining ones, in which case VIP(T + H k, V ) is an equality constrained problem, easier to solve than the original problem VIP(T, D). We look now at problems of the type VIP(T + H, V ), equivalent to finding a zero of T + H + N V, and, for the reasons just mentioned, we assume that V is an affine manifold, i.e. V = {x R n : Ax = b} with A R m n, b R n. In this case N V (x) = Im(A t ) for all x V, N V (x) = otherwise. H is assumed to be point-to-point and continuous. Finding a zero of T + H + N V is equivalent to finding x V such that H(x) T (x) + W (18) where W = Im(A t ). The proposal is to replace (18) by H(x) T ε (x) + W (19) for some ε > 0. We will prove that if T is uniformly monotone and x solves (18) then any point x close enough to x solves (19). 11

12 Lemma 2. Assume that T is maximal monotone and it satisfies any of the three hypotheses of Theorem 2. Let U dom T be open and take H: U R n continuous. Let V = {x R n : Ax = b} with A R m n, b R m and W = Im(A t ). Assume that U V. If x U is a solution of (18) then for all ε > 0 there exists η > 0 such that any x B(x, η) V solves (19). Proof: By Theorem 2 there exists δ such that T (B(x, δ)) T ε (x) for all x B(x, δ) (note that x (dom T ) 0 because x U). By Proposition 7, T 1 is continuous, so that there exists θ such that x T 1 (x) δ/2 for all x B(T (x), θ). By continuity of H, there exists ρ such that H(x) H(x) θ for all x B(x, ρ). Let η = min{ρ, δ 2 }. Take x B(x, η) V. Since η ρ we have H(x) H(x) θ. Let w = T 1 (T (x) + H(x) H(x)). Then T (w) T (x) = H(x) H(x). It follows that T (w) T (x) θ and therefore x w δ 2. Then x w x x + x w η + δ 2 δ. We have proved that w B(x, δ) and therefore T (w) T ε (x). So T (x) + H(x) H(x) T ε (x). Since x solves (18), T (x) + H(x) W. It follows that H(x) T ε (x) + W, i.e. x solves (19). Lemma 2 can be extended to the case in which V, instead of being an affine manifold, is of the form V = {x R n : f i (x) 0 (1 i m)} with f i convex and differentiable, but then the result must be modified in the following way: let I(x) = {i {1,... m} : f i (x) = 0}, V (x) = {x V : I(x) = I(x)}. Then any point in B(x, δ) V (x) solves (19). The reason is that when x V (x) then N V (x) and N V (x) (though not equal as in the case of the affine manifold) are close enough to each other. We have prefered to omit the details because in most relevant cases V will indeed be an affine manifold. A consequence of Lemma 2 for a particular algorithm will be presented in Section Bregman functions, quasi-fejér convergence and paramonotone operators This section contains the required material for the formulation and convergence analysis of the algorithm to be introduced in Section 4. Let C be a closed and convex subset of R n and C 0 its interior. Consider a convex real function g whose effective domain contains C and let D g : C C 0 R be defined as D g (x, y) = g(x) g(y) g(y) t (x y). (20) g is said to be a Bregman function (and D g the Bregman distance induced by g) if the following conditions hold: B1. g is continuously differentiable on C 0. B2. g is strictly convex and continuous on C. B3. For all δ R the partial level sets Γ(x, δ) = {y C 0 : D g (x, y) δ} are bounded for all x C. B4. If {y k } C 0 converges to y then D g (y, y k ) converges to 0. B5. If {x k } C and {y k } C 0 are sequences such that {x k } is bounded, lim k yk = y and lim D g(x k, y k ) = 0 then lim k k xk = y. C 0 is called the zone of g. These definitions originate in the results of [3]. It is easy to check that D g (x, y) 0 for all x C, y C 0 and D g (x, y) = 0 if and only if x = y. We 12

13 remark that B4 and B5 hold automatically when x k, y are in C 0, as a consequence of B1, B2 and B3, and so they need to be checked only at points in the boundary C of C. It has been proved in [6] that when C = R n a sufficient condition for a convex and differentiable function g to be a Bregman functions is lim x g(x) x =. It is easy to check that D g(, y) is strictly convex and continuous in C for all y C 0. By convention, we take D g (, ) = outside C C 0. Before presenting examples of Bregman functions, we introduce two subclasses to be used in the sequel. A Bregman function g is said to be zone coercive if: B6. For all y R n there exists x C 0 such that g(x) = y. A Bregman function g is said to be boundary coercive if: B7. If {x k } is a sequence contained in C 0 which converges to a point x in the boundary C of C, and y is any point in C 0 then lim k g(x k ), y x k =. Observe that condition B6 together with the strict convexity of g implies that the domain of g( ) is the set C 0. It has been proved in [5] that zone coerciveness implies boundary coerciveness. Example 4: C = R n, g(x) = x 2. In this case D g (x, y) = x y 2. Example 5: C = R n +, g(x) = n x j log x j, extended with continuity to R n + with the j=1 convention that 0 log 0 = 0. In this case D g (x, y) = Example 6: C = R n +, g(x) = n (x j log x j + y j x j ). y j j=1 n (x α j x β j ) with α 1, 0 < β < 1. For α = 2, j=1 β = 1 2 we get D g(x, y) = x y 2 + D g (x, y) = n j=1 1 2 y j ( x j y j ) 2. n j=1 1 2 y j ( x j y j ) 2, and for α = 1, β = 1 2 we get The Bregman functions of Examples 4 6 are all zone coercive, with the exception of Example 6 with α = 1, which is just boundary coercive. We will need in Section 4 a few properties of Bregman functions. Proposition 8. Let g be a Bregman function with zone C 0. Then i) D g (z, x) D g (z, y) D g (y, x) = g(x) g(y), y z for all z C, x, y C 0. ii) If g is either zone or boundary coercive then dom D g (, y) = C 0 for all y C 0. Proof: (i) follows immediately from (20). (ii) has been proved in [4, Lemma 1]. 13

14 Next we introduce the notion of quasi-fejér convergence. Let g be a Bregman function with zone C 0 and take Q C. A sequence {x k } C 0 is said to be quasi-fejér convergent to Q with respect to D g if and only if for all z Q there exists {ε k } R + such that ε k < and k=0 D g (z, x k+1 ) D g (z, x k ) + ε k. (21) The main properties of quasi-fejér convergence are summarized in the next proposition. Proposition 9. Assume that {x k } is quasi-fejér convergent to Q with respect to D g. Then i) {x k } is bounded. ii) For all z Q there exists a subsequence {x l k } of {x k } such that lim k (D g(z, x l k ) D g (z, x l k+1 )) = 0. iii) If a cluster point x of {x k } belongs to Q then x = lim k xk. Proof: k 1 i) By recurrence from (21) D g (z, x k ) D g (z, x 0 ) + ε i D g (z, x 0 ) + follows from B3 with δ = D g (z, x 0 ) + ε k. k=0 i=0 ε k. The result ii) If there exists a subsequence {x l k } such that D g (z, x lk+1 ) D g (z, x l k ) then, by (21), 0 D g (z, x lk+1 ) D g (z, x l k ) ε lk and the result follows because lim ε k = 0. Otherwise k there exists k such that 0 D g (z, x k+1 ) D g (z, x k ) for all k k. Then the sequence {D g (z, x k )} k k is nonincreasing and nonnegative, henceforth convergent, and therefore lim (D g(z, x k+1 ) D g (z, x k )) = 0, i.e. the whole sequence satisfies the requirement. k iii) Given any δ > 0 take ˆk such that ε k δ 2. Let {xl k } be a subsequence of {x k } which k= k converges to x. By B4 there exists k such that l k ˆk and D g (x, x l k) δ 2. Take any k l k. Then, by recurrence from (21) with z = x, D g (x, x k ) = D g (x, x l k) + k 1 i=l k ε i D g (x, x l k) + k=0 ε i δ 2 + δ 2 = δ. It follows that lim k D g(x, x k ) = 0. The result follows from B5 with y = x k = x, y k = x k. Quasi-Fejér convergence has been introduced in [7] and further studied in [10], where Proposition 9(i) and 9(iii) are proved for other type of generalized distances. For the case of g(x) = x 2, a stronger result can be found in [20], where Proposition 9(ii) is proved for 14 i= k

15 the whole sequence {x k }. Other applications of the notion of quasi-fejér convergence can be found in [16] and [17]. Next we deal with paramonotone operators, introduced in [5]. An operator T : R n P(R n ) is said to be paramonotone if it is monotone and additionally u v, x y = 0 with x, y R n, u T (x), v T (y) imply u T (y), v T (x). The main properties of paramonotone operators are summarized in the next proposition. Proposition 10. i) If T = f with f a convex function then T is paramonotone. ii) Assume that T is paramonotone in C and let z be a solution of VIP(T, C). Then y C is a solution of VIP(T, C) if and only if there exists v T (y) such that v, z y 0. iii) Let T : R n R n be a differentiable operator and J(x) its Jacobian matrix at x. If, for all x, J(x) + J(x) t is positive semidefinite and its rank is equal to the rank of J(x) then T is paramonotone. iv) If T 1 and T 2 are paramonotone then T 1 + T 2 is paramonotone. Proof: (i) Monotonicity of T is well known. Assume that u v, x y = 0 with x C, y C, u T (x), v T (y), and define f: C R as f(z) = f(z) + u, x z. Then f is convex and f(z) = f(z) u = {w u : w f(z)}. Taking w = u, we get that 0 f(x) and so x is an unrestricted minimizer of f. By hypothesis and definition of subgradients, f(x) f(y) u, x y = v, x y f(x) f(y), implying f(x) = f(y)+ u, x y = f(y). Since f(x) = f(x) by definition of f, we conclude that f(x) = f(y), so that y is also an unrestricted minimizer of f, i.e. 0 f(y). Therefore 0 = w u for some w f(y), which is equivalent to u f(y) = T (y). Reversing the roles of (x, u), (y, v) the same argument proves that v T (x) and the result is established. (ii) The only if part is immediate. We prove the if part. Assume that v, z y 0 for some v T (y) and some solution z of VIP(T, C). Pick u T (z) such that u, x z 0 for all x C. By monotonicity of T, 0 v, z y u, z y 0, implying that v, z y = u, z y = 0. Hence, v u, z y = 0 and by paramonotonicity of T we get u T (y). These two facts altogether imply, for all x C u, x y = u, x z + u, z y = u, x z 0 where we used the definition of solution in the rightmost inequality. It follows that y is a solution of VIP(T, C). (iii) See [11, Proposition 6]. (iv) Follows easily from the definition of paramonotonicity. 15

16 Finally we introduce a gap function for VIP(T, C), as defined in Section 1. Let q T,C (x) = sup{ v, x y : y C dom T, v T (y)}. (22) Several sufficient conditions for finiteness of q T,C in C are given in [4]. 4. An inexact proximal point method with Bregman distances for variational inequalities Take T : R n P(R n ) and C R n closed and convex. The variational inequality problem VIP(T, C) consists of finding z C such that there exists u T (z) satisfying u, x z 0 for all x C. From now on S(T, C) will denote the set of solutions of VIP(T, C). We make the following assumptions: A1. T = ˆT + N V with dom ˆT closed, (dom ˆT ) 0 C and V closed and convex (N V is the normal cone operator defined after Theorem 2). A2. ˆT is maximal monotone. A3. C 0 V. A4. S(T, C). Consider a Bregman function g with zone C 0 (i.e. conditions B1-B5 hold for g) satisfying A5. Either a) g is zone coercive, or b) g is boundary coercive and q T,C (x) < for all x C (with q T,C as in (22)). Take a sequence {λ k } R such that ˆλ λ k λ for some 0 < ˆλ λ and a sequence {ε k } R ++ such that ε k <. k=0 Let 1 D g (x, y) be the subdifferential of D g with respect to its first argument. The algorithm is defined as follows. Initialization: x 0 C 0 V. (23) Iterative Step: Given x k, define T k : R n P(R n ) as and take x k+1 such that T k = ˆT ε k + N V + λ k 1 D g (, x k ), (24) 0 T k (x k+1 ). (25) The exact algorithm considered in [4] is just (23)-(25) with ε k = 0 for all k. A convergence result similar to our next theorem can be found in [4] where an assumption A1 weaker than A1 is used: instead of the decomposition T = ˆT + N V with (dom T ) 0 C it is 16

17 asumed that dom T C 0 and that T is pseudomonotone (see [4]). Our slightly stronger hypothesis A1 gives rise to a considerable simplification in the proofs. We present next the convergence properties of the method. Theorem 3. Assume that A1-A5 hold. Let {x k } be the sequence generated by (23)-(25). Then i) The sequence {x k } is well defined and is contained in C 0 V. ii) The sequence {x k } is bounded. iii) If a cluster point x of {x k } solves VIP(T, C) then x = lim k xk. iv) If z S(T, C) then there exists a cluster point x of {x k } and u T (x ) such that u, z x 0. v) If ˆT is paramonotone then the sequence {x k } converges to a solution x of VIP(T, C). Proof: i) In [4, Theorems 1 and 2] it was proved, using essentially B2, A2 and A5, that the operator T k,y = ˆT + N V + λ k 1 D g (, y) has a unique zero for any y C 0. We proceed by induction. The result holds for k = 0 by A3 and (23). Assume that x k exists and belongs to C 0 V. By Propositions 1(i), 2(i) and A2, ˆT = ˆT 0 ˆT ε k, so that, using (24), T k,x k T k and then the unique zero of Tk,x k is a zero of T k, i.e. x k+1 exists. Since 0 T k (x k+1 ), x k+1 dom T k. From (24), dom T k = dom ˆT dom N V dom 1 D g (, x k ) = V C 0, using A1, dom N V = V, A5 and Proposition 8(ii). It follows that x k+1 V C 0 and the induction step is complete. ii) (iii) It follows easily from (20) that 1 D g (x, x k ) = g(x) g(x k ) (note that g(x k ) exists by (i) and B1). Then, using (24), we can rewrite (25) as λ k ( g(x k ) g(x k+1 )) ( ˆT ε k + N V )(x k+1 ). (26) Since N V is maximal monotone, using Proposition 1(ii) with ε 1 = ε k, ε 2 = 0, Proposition 2(i) and A1, we get ˆT ε k + N V = ˆT ε k + N 0 V ( ˆT + N V ) ε k = T ε k. (27) Let u k = λ k ( g(x k ) g(x k+1 )). From (26), (27) u k T ε k (x k+1 ). (28) Take z S(T, C) (nonempty by A4). Then there exists v T (z) such that, for all x C, By (1) and (28) v, x z 0. (29) ε k u k v, x k+1 z = u k, x k+1 z v, x k+1 z u k, x k+1 z = λ k g(x k ) g(x k+1 ), x k+1 z = λ k [D g (z, x k ) D g (z, x k+1 ) D g (x k+1, x k )] λ k [D g (z, x k ) D g (z, x k+1 )] (30) 17

18 using (29) in the first inequality, Proposition 8(i) in the last equality, and nonnegativity of D g in the last inequality. It follows from (30) that Since D g (z, x k+1 ) D g (z, x k ) + ε k λ k D g (z, x k ) + ε k λ. (31) ε k < and S(T, C) C we have proved that {x k } is quasi-fejér convergent k=0 to S(T, C) so that (ii) and (iii) follow from Proposition 9(i), 9(iii) respectively. iv) Fix z S(T, C). By Proposition 9(ii) there exists a sequence {x l k } of {x k } such that lim k (D g(z, x l k ) D g (z, x l k+1 )) = 0. By (30) Since λ k λ for all k and ε lk u l k, x l k+1 z λ lk (D g (z, x l k ) D g (z, x l k+1 )). (32) ε k <, we conclude from (32) that k=0 lim u l k, x lk+1 z = 0. (33) k In order to find cluster points of {u l k }, {x l k+1 } we must take care of some technical details. Since u k ( ˆT ε k + N V )(x k+1 ) by (26), we can write u k = u k + w k with u k ˆT ε k (x k+1 ), w k N V (x k+1 ). It follows easily from the definition of N V that w, y x 0 for all x, y V, w N V (x). Since x k+1 V by (i) and z S(T, C) dom T = dom ˆT dom N V dom N V = V, we conclude that w k, z x k+1 0, so that u k, z x k+1 = u k, z x k+1 + w k, z x k+1 u k, z x k+1. (34) Let Q = {x k k = 1, 2,... }. Since Q C 0 by (i), we get from A1 that Take ε such that ε k ε for all k. By Proposition 1(i) Q C (dom ˆT ) 0. (35) u k ˆT ε k (x k+1 ) ˆT ε (x k+1 ) ˆT ε (Q). (36) Since ˆT is maximal monotone by A2 and dom ˆT is closed by A1, we are within the hypotheses of Proposition 5, and so ˆT ε (Q) is bounded, i.e. {u k } is bounded. Since Q is bounded by (ii), we may assume without loss of generality (i.e. refining the subsequence {x l k } if necessary) that {u l k }, {x l k+1 } converge to some u, x respectively, so that we get from (33) and (34) u, z x 0. (37) 18

19 Since lim k ε k = 0, by (36), A2 and Propositions 1(iv) and 2(i) we get that u ˆT (x ). It is easy to check that 0 N V (x) for all x V, so that u = u + 0 ˆT (x ) + N V (x ) = ( ˆT + N V )(x ) = T (x ). (38) (iv) follows immediately from (37), (38). v) Since N V = I V, N V is paramonotone by Proposition 10(i). Since ˆT is paramonotone, T = ˆT + N V is paramonotone by Proposition 10(iv). By (iv) and Proposition 10(ii), {x k } has a cluster point x belonging to S(T, C). By (iii), x = lim k xk. The main advantage of (23)-(25) over the method in [4] (i.e. (23)-(25) with ε k = 0 for all k) is that now, instead of finding the unique zero of ˆT + NV + λ k 1 D g (, x k ), ε the k-th subproblem reduces to finding any zero of ˆT k + N V + λ k 1 D g (, x k ). In both cases the subproblems are generally easier than the original problem, because the term λ k 1 D g (, x k ) has a penalization effect, forcing the sequence {x k } to remain in C 0, so that the constraints in C are absent in the subproblems, which are either unconstrained (when V = R n ) so that they consist of finding a zero of the well behaved operator ˆT + λ k [ g( ) g(x k )], or constrained only by V, i.e. they reduce to VIP( ˆT + H k, V ), with H k (x) = λ k ( g(x) g(x k )). Since typically V will contain those constraints in D = C V which make D 0 empty, V will be in general an affine manifold, and then the equality constrained VIP( ˆT + H k, V ) will be considerably easier than VIP(T, C) = VIP( ˆT, C V ). But nevertheless, in the algorithm of [4] such easier subproblems must be exactly solved (and they have unique solutions). Now, with the help of Lemma 2, we will show that when V is an affine manifold, the solution sets of the subproblems of (23)-(25) contain a whole ball in V around the exact solution, corresponding to ε k = 0, under adequate regularity assumptions on ˆT. This means that the convergence properties of the algorithm are not affected by sufficiently small errors in the solution of the subproblems, and also that any explicit algorithm with feasible iterates guaranteeed to converge to a solution of a linear equality constrained variational inequality problem will find a solution of the subproblem after finitely many iterations. Corollary 4. Assume that A1-A5 hold, that V = {x R n : Ax = b} with A R m n, b R m and that, additionally, either i) ˆT = f for some convex function f, or ii) ˆT is uniformly monotone and point-to-point in (dom ˆT ) 0, or iii) ˆT is uniformly monotone and dom ˆT = R n. Let {x k } be the sequence generated by (23)-(25) and let ˆx k be the (unique) solution of 0 [ ˆT + N V + λ k 1 D g (, x k )](x). (39) Then there exists η k > 0 such that B(ˆx k, η k ) V is contained in the solution set of 0 T k (x) (40) 19

20 with T k as in (24), i.e. any element in B(ˆx k, η k ) V can be taken as x k+1 for algorithm (23)-(25). Proof: Let H k = λ k ( g(x) g(x k )) = λ k 1 D g (x, x k ). We check that we are within the hypotheses of Lemma 2 with U = C 0, H = H k, T = ˆT, x = ˆx k, ε = ε k and W = Im(A t ). H k is continuous in C 0 by B1, C 0 dom ˆT by A1, (39) is equivalent to (18) because N V (x) = Im(A t ) for all x V and (40) is equivalent to (19) by the same reason. Since (i)-(iii) are the hypotheses of Theorem 2, Lemma 2 holds and we can take η k = η, where η is as given by Lemma 2. References [1] Alber, Ya.I. On the regularization method for variational inequalities with nonsmooth unbounded operators in a Banach space. Applied Mathematics Letters 6,4 (1993) [2] Brézis, H. Opérateurs Monotones Maximaux et Semigroups de Contractions dans les Espaces de Hilbert. Mathematics Studies 5, North Holland, New York (1973). [3] Bregman, L.M. The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics 7,3 (1967) [4] Burachik, R.S., Iusem, A.N. A generalized proximal point algorithm for the variational inequality problem in a Hilbert space (submitted for publication). [5] Censor, Y., Iusem, A.N., Zenios, S.A. An interior point method with Bregman functions for the variational inequality problem with paramonotone operators (submitted for publication). [6] De Pierro, A.R., Iusem, A.N. A relaxed version of Bregman s method for convex programming. Journal of Optimization Theory and Applications 51 (1986) [7] Ermol ev, Yu.M. On the method of generalized stochastic gradients and quasi-fejér sequences. Cybernetics 5 (1969) [8] Hiriart-Urruty, J.-B., Lemarechal, C. Convex Analysis and Minimization Algorithms. Springer, Berlin (1993). [9] Iusem, A.N. An iterative algorithm for the variational inequality problem. Computational and Applied Mathematics 13 (1994) [10] Iusem, A.N., Svaiter, B.F., Teboulle, M. Entropy-like proximal methods in convex programming. Mathematics of Operations Research 19 (1994) [11] Iusem, A.N. On some properties of paramonotone operators (submitted for publication). [12] Kabbadj, S. Méthodes Proximales Entropiques. Thesis in Mathematics, Université de Montpellier, France (1994). [13] Kiwiel, K.C. Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics, Springer, Berlin 1133 (1985). 20

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