Semismooth Newton Methods for Operator Equations in Function Spaces

Transcription

1 Semismooth Newton Methods for Operator Equations in Function Spaces Michael Ulbrich April 2000, Revised October 2001 TR00-11 Accepted for Publication in SIAM Journal on Optimization Department of Computational and Applied Mathematics - MS134 Rice University 6100 Main Street Houston, Texas USA

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3 SEMISMOOTH NEWTON METHODS FOR OPERATOR EQUATIONS IN FUNCTION SPACES MICHAEL ULBRICH Abstract We develop a semismoothness concept for nonsmooth superposition operators in function spaces The considered class of operators includes NCP-function-based reformulations of infinite-dimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications Our results generalize semismoothness and α-order semismoothness from finite-dimensional spaces to a Banach space setting Hereby, a new infinite-dimensional generalized differential is used that is motivated by Qi s finite-dimensional C-subdifferential We apply these semismoothness results to develop a Newton-like method for nonsmooth operator equations and prove its local q-superlinear convergence to regular solutions If the underlying operator is α-order semismooth, convergence of q-order 1+α is proved We also establish the semismoothness of composite operators and develop corresponding chain rules The developed theory is accompanied by illustrating examples and by applications to nonlinear complementarity problems and a constrained optimal control problem Key words Newton-like methods, semismoothness, superposition operators, generalized differentials, nonlinear complementarity problems, superlinear convergence, optimal control problems AMS subject classifications 49M15,65K05,90C33,49J52,47J25,47H30 1 Introduction In this paper, we develop a semismoothness concept for nonsmooth operators in function spaces and establish q-superlinear convergence of a Newton-like method for semismooth operator equations Results on convergence with rate > 1 are also presented The class of operators we consider includes those obtained by NCP-function-based reformulations of nonlinear complementarity problems (NCP in function spaces These problems arise frequently in practice, eg, in form of first-order optimality conditions of constrained elliptic [58, 59], parabolic [60], and flow control problems [56, 58] As an illustrative example for the application to optimal control we will discuss the elliptic control problem (16 in detail The numerical results in [56, 58, 59] show that the semismooth Newton method developed in this paper solves constrained control problems very efficiently The notion of semismoothness was introduced by Mifflin [39] for real-valued functions defined on finite-dimensional spaces Qi [46] and Qi and Sun [48] extended semismoothness to mappings between finite-dimensional spaces and showed that, although the underlying mapping is in general nonsmooth, Newton s method can be generalized to semismooth equations and converges locally with q-superlinear rate to a regular solution [45, 46, 48] For related early approaches to nonsmooth Newton methods we refer to [36, 37, 43] In particular, Kummer [36, 37] has established q-superlinear convergence for a general, abstract class of nonsmooth Newton methods under conditions that include (11 Written in a form most convenient for our purposes, a mapping f : R k R l is called semismooth at x if f is Lipschitz near x, directionally differentiable at x, and if (11 max f(x + h f(x Mh = o ( h as h 0, M f(x+h where f denotes Clarke s generalized Jacobian [11] See Section 2 for details Further, if f is α-order semismooth, 0 <α 1, then the order in (11 can be improved to O ( h 1+α An important source of semismooth equations are reformulations of the nonlinear complementarity problem (NCP (12 y i 0, Z i (y 0, y i Z i (y =0, i =1,,k, Zentrum Mathematik, Technische Universität München, D München, Germany (mulbrich@matumde The author was supported by Deutsche Forschungsgemeinschaft (DFG grant Ul157/3-1 and by CRPC grant CCR

4 2 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES with continuously differentiable function Z : R k R k In this approach, which also can be applied to more general problems (mixed complementarity problem, MCP; variational inequality problem, VIP, an NCP-function [53], ie, a function φ : R 2 R with the property φ(x =0 x 1 0, x 2 0, x 1 x 2 =0, is applied component-wise to the NCP to rewrite it equivalently in the form (13 Φ(y =0, where Φ(y = ( φ(y 1,Z 1 (y,,φ(y k,z k (y T Frequently used NCP-functions are φ(x =min{x 1,x 2 } as well as the Fischer Burmeister function [20] (14 φ FB (x = x x2 2 x 1 x 2 Both are semismooth (of order 1, and thus the function Φ in (13 is semismooth, too Therefore, semismooth Newton methods can be applied to solve (13 The strong theoretical properties of this approach, its numerical potential, and extensions to more general problems (MCP, VIP have been extensively studied in recent years, see, eg, [15,17,18,32,33,57], and led to very efficient Newton-like methods, see, eg, [41] Although smooth NCP-functions can be constructed [38], they suffer from the fact that necessarily φ(0 = 0 must hold since the curve {x R 2 : φ(x =0} has a kink at x =0 As a consequence, the use of smooth NCP-functions requires a strict complementarity condition, whereas this can be avoided by working with nondifferentiable NCP-functions Since the introduction of the semismooth Fischer Burmeister function, many researchers agree that semismooth NCP-functions are a very powerful tool to develop efficient algorithms with strong theoretical properties The objective of this paper is to extend the notions of semismoothness and α-order semismoothness, respectively, to nonlinear superposition operators in function spaces, and to develop a corresponding superlinearly convergent Newton-like method Hereby, we are motivated by applications arising in mathematical modeling and optimal control, which often (see below can be cast as a pointwise bound-constrained variational inequality problem (VIP posed in function spaces As our main example we consider the following nonlinear complementarity problem: Find y L p (Ω such that (15 y 0, Z(y 0, yz(y =0, holds pointwise almost everywhere on Ω, where Ω R n is Lebesgue measurable with positive and finite measure, L p (Ω is the Lebesgue space of p-integrable functions, and the operator Z : L p (Ω L r (Ω, 1 r<p, is continuously Fréchet differentiable For the purpose of illustration, we now show how a particular optimal control problem can be converted to an NCP of the form (15 The problem we describe will serve as a model problem (chosen simple for convenience to which our theory and the developed Newton method is readily applicable Consider the following distributed optimal control problem of an elliptic partial differential equation with upper bounds on the control: (16a minimize w L 2 (Ω J(w def = 1 2 u(w u d 2 L 2 (Ω + λ 2 w w d 2 L 2 (Ω subject to w b on Ω, where u = u(w H0 1 (Ω (the usual Sobolev space is the weak solution of the uniformly elliptic state equation n ( u (16b a ij = w on Ω x i x j i,j=1

5 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES 3 We assume λ>0, a ij L (Ω, u d L 2 (Ω, and w d,b L (Ω Denoting by J(w L 2 (Ω the L 2 -Riesz representation of the gradient of J, it will be shown in Example 56 that w solves the control problem if and only if ȳ = b w solves the NCP (15 with Z(y = J(b y We will further discuss this problem in Example 56 and in Section 62 We stress that this problem is meant for the purpose of illustration, and so we decided to consider this particularly simple linear-quadratic control problem, which we hope is easily accessible to most readers For more advanced applications to the optimal control of nonlinear partial differential equations we refer the interested reader to [56, 58] In order to reformulate (15 as a nonsmooth operator equation, we use an NCP-function to rewrite the pointwise complementarity conditions in (15 as equations Doing this, (15 can be cast equivalently in form of the operator equation (17 Φ(y =0, where Φ(y(ω def = φ ( y(ω,z(y(ω, ω Ω In this paper, we consider superposition operators of the more general form (18 Ψ:Y L r (Ω, Ψ(y(ω =ψ ( F (y(ω, with mappings ψ : R m R and F : Y m i=1 Lr i (Ω, where 1 r r i <, Y is a real Banach space, and Ω R n is a bounded open domain Obviously, choosing Y = L p (Ω, r 1 = r 2 = r, m =2, ψ = φ, and F : y Y ( y,z(y,wehaveψ=φwith Φ and Ψ as in (17 and (18, respectively, so that reformulated NCPs are included as special cases in our analysis Essentially, our working assumptions are that ψ is Lipschitz continuous and semismooth, and that F is continuously Fréchet differentiable The detailed assumptions are given below The main result of this paper is a semismoothness-like estimate of the form (19 sup Ψ(y + s Ψ(y Ms L r = o( s Y as s 0 in Y M Ψ(y+s We also give conditions under which the remainder term in (19 is of the order O( s 1+α Y, 0 <α 1 In this case we call Ψ α-order semismooth The multifunction (ie, set-valued mapping Ψ:Y L(Y,L r denotes an appropriate vector-valued generalized differential of Ψ, which is related to, and motivated by Qi s finite-dimensional C-subdifferential [47] The estimate (19 generalizes (11 to the function space setting We will not require that Ψ be directionally differentiable, because this is not needed in the analysis of Newton s method We remark that several authors [22, 36, 40, 63] have studied conditions of the form (19 in finite dimensions independently of the papers [46] and [48] Based on (19, we develop a locally q-superlinearly convergent Newton method for the nonsmooth operator equation (110 Ψ(y =0 Moreover, in the case where Ψ is α-order semismooth we prove convergence with q-rate 1+α In analogy to BD-regularity assumptions for finite-dimensional semismooth Newton methods, we impose a regularity condition on the elements of the generalized differential Further, as was already observed earlier in the context of related local convergence analyses in function space [34, 60], we have to incorporate a smoothing step to overcome the nonequivalence of norms We also will provide an example showing that this smoothing step can be indispensable Recently, a different semismoothness concept for operator equations was proposed by Chen, Nashed and Qi [10] We point out that our approach differs significantly from the one in [10] There, the notion of a slanting function is introduced and a generalized derivative, the slant derivative, is obtained as the collection of all limits of the slanting function as y k y

6 4 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES Semismoothness is then defined by imposing appropriate conditions on the approximation properties of the slanting function and the slant derivative Although the differentiability properties of superposition operators with smooth ψ are well investigated, see, eg, the expositions [6] and [7], this is not the case for nonsmooth functions ψ Further, even if ψ is smooth, for operator equations of the form (110 the availability of local convergence results for Newton-like methods appears to be very limited As an important application, and to illustrate our results, we discuss reformulations (17 of the nonlinear complementarity problem (15 Furthermore, we show how the constrained elliptic control problem (16 can be converted to an equivalent NCP that meets all our assumptions There are also close connections between the NCP-function approach and non-interior path-following methods for NCPs [9], which recently were introduced and analyzed in finite dimensions Hereby, the NCP-function φ is embedded in a class of smooth perturbations φ σ, where σ 0 is a parameter For σ>0 the function φ σ is smooth, whereas φ 0 = φ For the Fischer-Burmeister function φ FB, eg, the functions φ σ can be obtained by adding the term σ under the square root The main idea of these methods, transcribed to our setting, consists in following the trajectory of solutions to the corresponding perturbed operator equations Φ σ (y =0as σ 0 Usually, corrector steps are computed by Newton s method In the asymptotic phase σ 0 the behavior of Newton s method on the unperturbed equation plays a key role in achieving fast local convergence We therefore believe that the results presented in this paper will also be helpful to investigate path-following methods in a function space setting We emphasize that the number of applications fitting in our framework is huge, in particular those involving complementarity, see [13, 16, 19, 24, 26, 35, 42, 44] Many of these applications arise from infinite-dimensional variational inequalities that model systems being continuous in time and/or space [13, 16, 24, 35, 42], and therefore are posed in function spaces Hence, the development and analysis of efficient abstract algorithms for the solution of the infinite-dimensional problem (15 is very desirable in order to derive robust, efficient, and mesh-independent methods for the solution of the discretized problem The nonsmooth Newton method developed in this paper is directly applicable to NCP-function-based reformulations of the NCP (15 and can therefore be seen as a generalization of semismooth Newton methods for finite-dimensional NCPs For the development of a semismoothness concept we have to choose an appropriate vector-valued generalized differential for the operator Ψ Although the available literature on generalized differentials and subdifferentials is mainly focused on real-valued functions, see, eg, [8, 11, 12, 51] and the references therein, several authors have proposed and analyzed generalized differentials for nonlinear operators between infinite-dimensional spaces [14, 23, 28,49,54] In our approach, we work with a generalized differential that exploits the structure of Ψ Roughly speaking, our general guidance hereby is to transcribe, at least formally, componentwise operations in R k to pointwise operations in function spaces To sketch the idea, note that the finite-dimensional analogue of the operator Ψ is the mapping Ψ f : R k R l, Ψ f j(x =ψ ( F j (x, j =1,,l with ψ as above and C 1 -mappings F j : R k R m We have the correspondences ω Ω j {1,,l}, y Y x R k, and F (y(ω F j (x Componentwise application of the chain rule for Clarke s generalized gradient [11] shows that the C-subdifferential of Ψ f consists of matrices M R l k having rows of the form M j = m i=1 d j i (F j i (x, with d j ψ ( F j (x

7 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES 5 Note that the collection of all these matrices M can be an overestimate of the C-subdifferential, since the chain rule asserts only that [ ψ ( F j (x ] ψ ( F j (x (F j (x Carrying out the same construction for Ψ in a purely formal manner suggests to choose a generalized differential for Ψ consisting of operators of the form v Y m i=1 d i (F i (xv, with (d 1,,d m (ω ψ ( F (y(ω ae on Ω, where the inclusion on the right is meant in the sense of measurable selections One advantage of this approach, which motivates our choice of the generalized differential Ψ, is that it consists of relatively concrete objects as compared to those investigated in, eg, [14, 23, 28, 49, 54], which necessarily are more abstract since they are not restricted to a particular structure of the underlying operator It is not the objective of this paper to investigate the connections between the generalized differential Ψ and other generalized differentials There are close relationships, but we leave it as a topic for future research Here, we concentrate on the development of a semismoothness concept based on Ψ, a related nonsmooth Newton s method, and the relations to the respective finite-dimensional analogues As already mentioned, the literature on Newton-like methods for the solution of nonlinear complementarity problems or, closely related, bound-constrained optimization problems posed in function spaces is very limited Hereby, we call an iteration Newton-like if each iteration essentially requires the solution of a linear operator equation We point out that in this sense sequential quadratic programming (SQP methods for problems involving inequality constraints [1 5, 27, 55] are not Newton-like, since each iteration requires the solution of a quadratic programming problem (or, put differently, a linearized generalized equation which is in general significantly more expensive than solving a linear operator equation Therefore, instead of applying the methods considered in this paper directly to the nonlinear problem, they also could be of interest as subproblem solvers for SQP methods Probably the investigations closest related to ours are the analysis of Bertsekas projected Newton method by Kelley and Sachs [34], and the investigation of affine-scaling interiorpoint Newton methods by Ulbrich and Ulbrich [60] Both papers deal with bound-constrained minimization problems in function spaces and establish the local q-superlinear convergence of their respective Newton-like methods In both approaches the convergence results are obtained by estimating directly the remainder terms appearing in the analysis of the Newton iteration Hereby, specific properties of the solution are exploited, and a strict complementarity condition is assumed in both papers We develop our results for the general problem class (110 and derive the applicability to nonlinear complementarity problems as a simple, but important special case In the context of NCPs and optimization, we do not have to assume any strict complementarity condition Further, we organize our analysis of Newton s methods by decomposing it in two parts: First, we develop a semismoothness result that replaces differentiability in ordinary Newton methods Second, an invertibility condition on the members of the generalized differential is introduced This regularity condition can be verified conveniently by using the sufficient conditions that we recently developed in [58, 59] In Section 2 we review some concepts of finite-dimensional nonsmooth analysis that are important in our context, in particular generalized differentials and semismoothness Our working assumptions are stated in Section 3 In Section 4 we introduce the generalized differential Ψ and investigate some of its properties In Section 5 a semismoothness and α-order semismoothness concept for the operator Ψ is proposed and studied in detail The results are illustrated by applications to nonlinear complementarity problems In particular, we demonstrate the necessity of our assumptions by several (counter- examples In Section 6 we propose a Newton-like method for the solution of the nonsmooth operator equation (110 and use our semismoothness results to establish its q-superlinear convergence In the case of a α-order semismooth operator Ψ we prove convergence of q-order 1 + α Applications to

8 6 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES NCPs are provided as illustrating examples and the computation of smoothing steps is discussed Furthermore, we consider the application of the semismooth Newton method to the elliptic control problem (16 and address its discretization In Section 7 we show that under appropriate assumptions the composition of semismooth operators is again semismooth and develop two chain rules Finally, in Section 8, we establish some further properties of our generalized differential Notations Given a Banach space Y, we denote by Y its norm, by B Y its open unit ball, and by B Y its closed unit ball; in the special case Y =(R n, p, we prefer to write Bp n and B p n, respectively On a product space i Y i, we choose y Πi Y i = i y Y i as norm L(Y,Z denotes the Banach space of bounded linear operators from the Banach space Y to the Banach space Z, equipped with the operator norm Y,Z By v, w Ω we denote the dual pairing between v L p (Ω and w L p (Ω, 1/p +1/p =1 The indicator function of a measurable set Q Ω, taking the value one on Q and zero on its complement Q c =Ω\ Q,is denoted by 1 Q We write µ for the Lebesgue measure on R n Given a function w L (Ω and an operator A L(Y,L p (Ω, we define the operator w A L(Y,L p (Ω that takes y Y to the function ω Ω w(ω(ay(ω The Fréchet derivative of an operator H is denoted by H For convenience, we will write i and i instead of m i=1 and m i=1 2 Generalized differentials and semismoothness in finite dimensions We begin with an overview of the semismoothness concept in finite dimensions Let the vector-valued function f : R k R l be given We first collect some notions from nonsmooth analysis Assume that f is locally Lipschitz continuous According to Rademacher s theorem, the set U f R k of all points x at which f fails to be differentiable is a Lebesgue null set Hereby, the fact that f is a mapping between finite-dimensional spaces is crucial Using this, generalized Jacobians can be constructed: DEFINITION 21 Let f be locally Lipschitz We define the following generalized Jacobians of f at x: (a The Bouligand (B- subdifferential: B f(x def = { M R l k : (x j R k \ U f : x j x, f (x j M }, where f denotes the Jacobian of f (b Clarke s generalized Jacobian is the convex hull of B f(x: f(x def = co B f(x (c Qi s C-subdifferential: C f(x def = f 1 (x f l (x These generalized differentials induce multifunctions B f, f, C f : R k R l k They have the following properties: (a B f, f, and C f are nonempty-, and compact-valued Moreover, f and C f are convex-valued (b The multifunctions B f, f, and C f are upper semicontinuous (see Definition A2 or [11, p 29] (c B f(x f(x C f(x for all x Based on Clarke s generalized Jacobian, Qi [46] and Qi and Sun [48] introduced the following notion of semismoothness: DEFINITION 22 f is semismooth at x R k if it is locally Lipschitz and, for all h R k, the limit lim Mh M f(x+th h h,t 0 +

9 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES 7 exists and is finite The following characterization, however, is more appropriate for our purposes: PROPOSITION 23 Let f be locally Lipschitz Then f is semismooth at x if and only if f is directionally differentiable at x and (21 max f(x + h f(x Mh 2 = o( h 2 as h 0 M f(x+h Proof In [48, Thm 23] it is shown that the locally Lipschitz continuous function f is semismooth at x if and only if f is directionally differentiable at x and (22 max Mh f (x, h 2 = o( h 2 as h 0 M f(x+h Furthermore, since f is locally Lipschitz continuous on the finite-dimensional space R k, directional differentiability implies Bouligand- (B- differentiability [52]: (23 f(x + h f(x f (x, h 2 = o( h 2 as h 0 It is now straightforward to see that under (23 the conditions (21 and (22 are equivalent DEFINITION 24 f is α-order semismooth, 0 < α 1, atx R k if it is locally Lipschitz, directionally differentiable at x, and if max M f(x+h Mh f (x, h 2 = O ( h 1+α 2 as h 0 The following consequence of α-order semismoothness will be important: PROPOSITION 25 ( [21, Lem 2, Lem 17] Let f be α-order semismooth at x, 0 < α 1 Then (24 (25 max f(x + h f(x Mh 2 = O( h 1+α 2 as h 0, M f(x+h f(x + h f(x f (x, h 2 = O ( h 1+α 2 as h 0 It is obvious that useful semismoothness concepts can also be obtained by replacing f by other suitable generalized derivatives This was investigated in a general framework by Jeyakumar [29, 30] and by Xu [62, 63] Here, we only sketch Jeyakumar s approach, in which he introduced the concept of f-semismoothness, where f is an approximate Jacobian [31] For the definition of approximate Jacobians we refer to [31]; in the sequel, it is sufficient to know that an approximate Jacobian of f : R k R l is a closed-valued multifunctions f : R k R l k and that B f, f, and C f are approximate Jacobians DEFINITION 26 Let f : R k R l be continuous and let be given an approximate Jacobian f of f (a The function f is called weakly f-semismooth at x if (26 sup f(x + h f(x Mh 2 = o( h 2 as h 0 M co f(x+h (b The function f is f-semismooth at x if (i f is B-differentiable at x (eg, locally Lipschitz near x and directionally differentiable at x, see [52], and (ii f is weakly f-semismooth at x

10 8 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES Note that f-semismoothness coincides with semismoothness Obviously, we can define weak f-semismoothness of order α by requiring the order O ( h 1+α 2 in (26 Finally, we consider a Newton-like method for the solution of the nonsmooth equation (27 f(x =0, where f : R k R k is weakly f-semismooth or weakly f-semismooth of the order α, respectively, at the solution x For this system of equations, Newton-like methods were developed that converge locally q-superlinearly [29, 45, 46, 48], see also [36, 37] A representative result is the following PROPOSITION 27 Denote by x R k a solution of (27 and let the initial point x 0 R k be given Consider the following Newton-like iteration: For j =0, 1, 2,as long as f(x j 0: Choose M j f(x j and compute x j+1 = x j + s j, where M j s j = f(x j Assume that (a f is weakly f-semismooth (or weakly f-semismooth of the order α at x (b There exist η>0and C>0such that, for all x x + ηb2 k, every M f(x is nonsingular with M 1 2 C (Regularity assumption Then there exists δ>0such that for all x 0 x + δb2 k the above iteration either terminates with x j = x or generates a sequence (x j that converges q-superlinearly (or with q-order 1+α to x Proof As long as x j x + ηb2 k, the iteration is well defined by (b Setting e j = x j x and using f( x =0,wehave M j e j+1 = M j s j + M j e j = f(x j +M j e j = f( x f( x + e j +M j e j This, (a, and (b yield (28 e j+1 2 M 1 j 2 f( x + e j f( x M j e j 2 = o( e j 2 as x j x By (a we can choose δ (0,η] so small that (29 f( x + h f( x Mh 2 h 2 for all M f( x + h and all h δb2 k 2C Note that this trivially holds for h =0 Hence, for all x j x + δb2 k with x j x, we have e j+1 2 e j 2 /2 by (28, and thus x j+1 x +( e j 2 /2B2 k x +(δ/2bk 2 Inductively, we conclude that for all x 0 x + δb2 k the algorithm is well defined and either terminates finitely or generates a sequence (x j converging to x In the case of finite termination we have f(x j = 0 and, by (29 and the choice of δ, we see that, for any M co f( x + e j, e j 2 /2 C f(x j f( x Me j 2 M 1 2 Me j 2 e j 2, hence x j = x On the other hand, if the algorithm generates an infinite sequence x j x then we see from (28 that the rate of convergence is q-superlinear If f is weakly f- semismooth of order α at x, then we can improve the order in (28 to O( e j 1+α 2 and obtain convergence with q-rate 1+α REMARK 28 In many cases, the approximate Jacobian is upper semicontinuous and compact-valued, in particular if B f, f, or C f are used Then it is easy to show that the regularity condition 27 (b is already satisfied if all M f( x are nonsingular

11 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES 9 3 Assumptions In the rest of the paper, we will impose the following assumptions on F and ψ: ASSUMPTION 31 There are 1 r r i <q i, 1 i m, such that (a The operator F : Y i Lr i (Ω is continuously Fréchet differentiable (b The mapping y Y F (y i Lq i (Ω is locally Lipschitz continuous, ie, for all y Y there exists an open neighborhood U = U(y and a constant L F = L F (U such that F i(y 1 F i (y 2 i L q i L F y 1 y 2 Y for all y 1,y 2 U (c The function ψ : R m R is Lipschitz continuous of rank L ψ > 0, ie, ψ(x 1 ψ(x 2 L ψ x 1 x 2 1 for all x 1,x 2 R m, (d ψ is semismooth REMARK 32 Since by assumption the domain Ω is bounded, we have the continuous embedding L q (Ω L p (Ω whenever 1 p q REMARK 33 Note that in Assumption 31 the only difference between the operators in (a and (b is the topology of the range space As mentioned in Remark 32, the L q i -norms are stronger than the corresponding L r i -norms For semismoothness of order > 0 we will strengthen the Assumptions 31 as follows: ASSUMPTION 34 As Assumption 31, but with (a and (d replaced with: There exists α (0, 1] such that (a The operator F : Y i Lr i (Ω is α-order Hölder continuously Fréchet differentiable (d ψ is α-order semismooth Note that for the special case Y = i Lq i (Ω and F = id Y we have Ψ:y Y ψ(y, and it is easily seen that the Assumptions 31 or 34, respectively, reduce to (c and (d or (c and (d, respectively Under the Assumptions 31, the operator Ψ defined in (18 is well defined and locally Lipschitz continuous PROPOSITION 35 Let the Assumptions 31 hold Then for all 1 q q i, 1 i m, and thus in particular for q = r, the operator Ψ defined in (18 maps Y locally Lipschitz continuous into L q (Ω Proof Using Lemma A1, we first prove Ψ(Y L q (Ω, which follows from Ψ(y L q = ψ ( F (y L q ψ(0 L q + ψ ( F (y ψ(0 L q c q, (Ω ψ(0 + L ψ i F i(y L q c q, (Ω ψ(0 + L ψ i c q,q i (Ω F i (y L q i To establish the local Lipschitz continuity, denote by L F the local Lipschitz constant in As-

12 10 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES sumption 31 (b on the set U and let y 1,y 2 U be arbitrary Then, again by Lemma A1, Ψ(y 1 Ψ(y 2 L q L ψ i F i(y 1 F i (y 2 L q L ψ i c q,q i (Ω F i (y 1 F i (y 2 L q i ( L ψ L F max c q,q i (Ω y 1 y 2 Y 1 i m 4 An infinite-dimensional generalized differential For the development of a semismoothness concept for the operator Ψ defined in (18 we have to choose an appropriate generalized differential As we already mentioned in the introduction, our aim is to work with a differential that is as closely connected to finite dimensional generalized Jacobians as possible Hence, we will propose a generalized differential Ψ in such a way that its natural finite-dimensional discretization contains Qi s C-subdifferential, see Section 62 Our construction is motivated by a formal pointwise application of the chain rule In fact, suppose for the moment that the operator y Y F (y C( Ω m is strictly differentiable, where C( Ω denotes the space of continuous functions equipped with the max-norm Then for fixed ω Ω the function f : y F (y(ω is strictly differentiable with derivative f (y L(Y,R m, f (y :v ( F (yv (ω The chain rule for generalized gradients [11, Thm 2310] applied to the real-valued mapping y Ψ(y(ω =ψ ( f(y yields (41 ( Ψ(y(ω ψ ( f(y { g,v = f (y = g Y i d i(ω ( F i (yv } (ω, d(ω ψ ( F (y(ω Furthermore, we can replace by = ifψ or ψ is regular (eg, if ψ is convex or concave or if the linear operator f (y is onto, see [11, Thm 2310] Following the above motivation, and returning to the general setting of Assumption 31, we define the generalized differential Ψ(y in such a way that for all M Ψ(y, the linear form v (Mv(ω is an element of the right hand side in (41: DEFINITION 41 (Generalized differential Ψ Let the Assumptions 31 hold For Ψ as defined in (18 we define the generalized differential Ψ:Y L(Y,L r, (42 Ψ(y def = { M L(Y,L r M : v i d i (F i (yv, d measurable selection of ψ ( F (y REMARK 42 The superscript is chosen to indicate that this generalized differential is designed for superposition operators The generalized differential Ψ(y is nonempty To show this, we first prove: LEMMA 43 Let the Assumption 31 (a hold and let d L (Ω m be arbitrary Then the operator M : v Y d i (F i (yv i is an element of L(Y,L r and (43 M Y,L r i c r,r i (Ω d i L F i (y Y,L r i }

13 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES 11 Proof By Assumption 31 (a and Lemma A1 Mv L r = d i (F i (yv L d i i r i L F i (yv L r ( c r,r i (Ω d i i L F i (y Y,L r i v Y for all v Y, which shows that (43 holds and M L(Y,L r In a next step, we show that the multifunction ψ ( F (y : ω Ω ψ ( F (y(ω R m is measurable (see Definition A3 or [50, p 160] LEMMA 44 Any closed-valued, upper semicontinuous multifunction Γ:R k R l is Borel measurable Proof Let C R l be compact We show that Γ 1 (C is closed To this end, let x k Γ 1 (C be arbitrary with x k x Then there exist z k Γ(x k C, and, due to the compactness of C, we achieve by transition to a subsequence that z k z C Since x k x, upper semicontinuity yields that there exist ẑ k Γ(x with (z k ẑ k 0 and thus ẑ k z Therefore, since Γ(x is closed, we obtain z Γ(x C Hence, x Γ 1 (C, which proves that Γ 1 (C is closed and therefore a Borel set COROLLARY 45 The multifunction ψ ( F (y :Ω R is measurable Proof By Lemma 44, the compact-valued and upper semicontinuous multifunction ψ is Borel measurable Now, for all closed sets C R m, we have, setting u = F (y i Lr i (Ω, ψ ( F (y 1 (C =u 1 ( ψ 1 (C This set is measurable, since ψ 1 (C is a Borel set and u is a (class of equivalent measurable function(s The next result is a direct consequence of Lipschitz continuity, see [11, 212] LEMMA 46 Under Assumption 31 (c there holds ψ(x [ L ψ,l ψ ] m for all x R m Combining this with Corollary 45 yields: LEMMA 47 Let the Assumptions 31 hold Then for all y Y, the set (44 K(y = { d :Ω R m : d measurable selection of ψ ( F (y } is a nonempty subset of L ψ Bm L L (Ω m Proof By the Theorem on Measurable Selections [50, Cor 1C] and Corollary 45, ψ ( F (y admits at least one measurable selection d :Ω R m, ie, d(ω ψ ( F (y(ω ae on Ω From Lemma 46 follows d L ψ Bm L We now can prove: PROPOSITION 48 Under the Assumptions 31, for all y Y the generalized differential Ψ(y is nonempty and bounded in L(Y,L r Proof Lemma 47 ensures that there exist measurable selections d of ψ ( F (y and that all these d are contained in L ψ Bm L Hence, Lemma 43 shows that M : v d i (F i (yv i is in L(Y,L r The boundedness of Ψ(y follows from (43

14 12 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES We now have everything at hand to introduce a semismoothness concept that is based on the generalized differential Ψ We postpone the investigation of further properties of Ψ to the Sections 7 and 8 There, we will establish chain rules, the convex-valuedness, weak compact-valuedness, and the weak graph closedness of Ψ 5 Semismoothness in function spaces In this section, we develop a semismoothness concept for the operator Ψ defined in (18 Our notion of semismoothness is similar to Jeyakumar s weak semismoothness in Definition 26 (a In place of the finite-dimensional approximate Jacobian we work with the generalized differential Ψ Since we will show in Theorem 81 that Ψ is convex and closed (even compact in the weak operator topology, there is no need of taking the closed convex hull of Ψ as is done in (26 DEFINITION 51 The operator Ψ is semismooth at y Y if (51 sup Ψ(y + s Ψ(y Ms L r = o( s Y as s 0 in Y M Ψ(y+s Ψ is α-order semismooth, 0 <α 1,aty Y if (52 sup Ψ(y + s Ψ(y Ms L r = O ( s 1+α Y as s 0 in Y M Ψ(y+s This definition is easily extended to general operators between Banach spaces Of course, an appropriate generalized differential must be available In this paper, we only deal with the superposition operator Ψ and thus we dispense with a more general definition of semismoothness In the following main theorem we establish the semismoothness and the β-order semismoothness, respectively, of the operator Ψ THEOREM 52 (a Under the Assumptions 31, the operator Ψ is semismooth (b Let the Assumptions 34 hold Assume that there exists γ>0such that the set { ( Ω ε = ω : max ρ ( F (y(ω,h } ε α h 1+α 1 > 0, ε > 0, h 1 ε with the residual function ρ : R m R m R given by ρ(x, h = has the following decrease property: max ψ(x + h ψ(x z T h, z T ψ(x+h (53 µ(ω ε =O(ε γ as ε 0 +, Then the operator Ψ is β-order semismooth at y with { } γν αγν β = min,, where 1+γ/q 0 α + γν (54 q 0 = min q i, 1 i m ν = q 0 r q 0 r if q 0 <, ν = 1 r if q 0 = The proof of this theorem will be presented in Section 51 REMARK 53 Condition 53 requires the measurability of the set Ω ε, which will be verified in the proof We also remark that the α-order semismoothness of ψ implies µ(ω ε 0 as ε 0, see the discussion after Remark 54 REMARK 54 As we will see in Lemma 58, it would be sufficient to require only the β-order Hölder continuity of F in Assumption 34 (a with β α as defined in (54

15 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES 13 It might be helpful to give an explanation of the abstract condition (53 here For convenient notation, let x = F (y(ω Due to the α-order semismoothness of ψ provided by Assumption 34, we have ρ ( x, h = O ( h 1+α 1 as h 0, see Proposition 25 In essence, Ω ε is the set of all ω Ω where there exists h ε B 1 m for which this asymptotic behavior is not yet observed, because the remainder term ρ ( x, h exceeds h 1+α 1 by a factor of at least ε α, which grows infinitely as ε 0 From the continuity of the Lebesgue measure it is clear that µ(ω ε 0 as ε 0 The decrease condition (53 essentially states that the measure of the set Ω ε where F (y takes bad values, ie, values at which the radius of small residual is very small, decreases with the rate ε γ The following Example 55 demonstrates the applicability of Theorem 52 to nonlinear complementarity problems It also provides a very concrete interpretation of condition (53 EXAMPLE 55 (Application to NCPs The reformulation of nonlinear complementarity problems (15 in the form (17 leads to an important special case of the operator equations (110 under consideration Let the operator Z : L p (Ω L r (Ω, 1 r<p,begiven and consider the NCP (15, which we restate for convenience: (55 y 0, Z(y 0, yz(y =0 In Section 1 we showed that we can apply an NCP-function φ to transform (55 to the equivalent operator equation (56 Φ(y =0, where Φ(y(ω =φ ( y(ω,z(y(ω, ω Ω We now view the operator Φ as a special case of the more general class of operators Ψ defined in (18 and interpret Assumptions 31 and 34 in this context To this end, we choose Y = L p (Ω, set r 1 = r 2 = r, and define F : y Y ( y,z(y L r 1 (Ω L r 2 (Ω Then (56 is equivalent to (110 with ψ = φ Assume that (a The operator Z : L p (Ω L r (Ω is continuously Fréchet differentiable (b There is q (r, ] such that Z : L p (Ω L q (Ω is locally Lipschitz continuous (c φ is Lipschitz continuous (d φ is semismooth Then the Assumptions 31 are satisfied with q 1 = p and q 2 = q In fact, (a and the continuous embedding L p (Ω L r (Ω imply 31 (a Further, (b and the Lipschitz continuity of the identity u L p (Ω u L p (Ω yield 31(b Finally, (c,(d imply 31 (c,(d Therefore, we can apply Theorem 52 and obtain that Φ is semismooth: (57 sup Φ(y + s Φ(y Ms L r = o( s L p as s 0 in L p (Ω M Φ(y+s Further, we have for all M Φ(u and v Y (58 Mv = d 1 v + d 2 (Z (yv, where d L (Ω 2 is a measurable selection of φ ( y,z(y In the next Example 56 we will show that the optimal control problem (16 can be converted to an equivalent NCP for which the above assumptions (a, (b are satisfied In the rest of this example we focus on semismoothness of order β>0 As above we see that Assumption 34 holds if instead of (a and (d we require (a The operator Z : L p (Ω L r (Ω is α-hölder continuously Fréchet differentiable (d φ is α-order semismooth

16 14 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES If also condition (53 is satisfied, we can apply Theorem 52 to derive the β-order semismoothness of Φ Once we have chosen a particular NCP-function, condition (53 can be made very concrete We discuss this for the Fischer Burmeister function φ = φ FB, which is Lipschitz continuous and 1-order semismooth, and thus satisfies Assumptions 34 (c and (d with α =1 Further, this function is C on R 2 \{0} with derivatives φ(x = x x 2 ( 1 1, 2 φ(x = 1 x 3 2 ( x 2 2 x 1 x 2 x 1 x 2 x 2 1 The eigenvalues of 2 φ(x are 0 and x 1 2 In particular, we see that 2 φ(x 2 = x 1 2 explodes as x 0 If0 / [x, x + h], then Taylor expansion of φ(x about x + h yields with appropriate τ [0, 1] ρ(x, h = φ(x + h φ(x φ(x + h T h = 1 2 ht 2 φ(x + τhh h x + τh 2 Further, ρ(0,h=0, ρ(x, 0 = 0 Our aim is to show that (53 is equivalent to the condition (59 µ ({0 < F (y 1 <ε} =O(ε γ as ε 0 Obviously, this follows easily when we have established the following relation: { (510 {0 < F (y 1 <ε} Ω ε 0 < F (y 1 < ( 1+2 1/2 } ε To show the first inclusion in (510, let ω be such that x = F (y(ω satisfies 0 < x 1 <ε and choose h = tx, where t (1, 2 is such that h 1 ε Then a straightforward calculation yields ρ(x, h =2 x x 1 = h t 1 > h 1 ε 1 h 2 1 This implies ω Ω ε and thus proves the first inclusion To show the second inclusion in (510, let u = F (y Ifu(ω =0then certainly ω/ Ω ε, since then ρ ( u(ω, 0 If on the other hand u(ω 1 ( 1+2 1/2 ε then we have for all h ε B 1 2 ρ ( u(ω,h h u(ω+τh 2 h u(ω+τh 1 ε 1 h 2 1, and thus ω/ Ω ε Having established the equivalence of (53 and (59, the meaning of (53 becomes apparent: The set {0 < F (y 1 <ε} on which the decrease rate in measure is assumed is the set of all ω where strict complementarity holds, but is less than ε, ie, 0 < y(ω + Z(y(ω < ε In a neighborhood of these points the curvature of φ is very large since 2 φ is big This requires that F (y + s(ω F (y(ω must be very small in order to have a sufficiently small residual ρ ( F (y(ω,f(y + s(ω F (y(ω We stress that a violation of strict complementarity, ie, y(ω =Z(y(ω =0does not cause any problems since then ρ(f (y(ω, =ρ(0, 0 In the next example, we return to the control problem (16 EXAMPLE 56 (Application to a control problem We consider the constrained elliptic control problem (16 and show that it is equivalent to an NCP satisfying the conditions (a and (b derived in the previous Example 55 Further, we establish additional results that will

17 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES 15 be useful in Section 62 where we describe how the developed semismooth Newton method can be applied to solve the control problem Denote by A L(H0 1,H 1 the linear operator on the left hand side of (16b Due to the uniform ellipticity assumption, it is well known that A is a homeomorphism, so that, using the continuous embedding L 2 (Ω H 1 (Ω = H0 1(Ω, the control-to-state mapping w L 2 (Ω u(w =A 1 w H0 1 (Ω is continuous linear and thus smooth with Fréchet derivative u (w :v L 2 (Ω A 1 v H0 1(Ω Therefore, denoting by J(w L2 (Ω the L 2 -Riesz representation of the gradient of J,wehave J(w =(A 1 (A 1 w u d +λ(w w d The first-order necessary (and here also sufficient optimality conditions for (16a result in the pointwise complementarity system (511 w b, J(w 0, (w b J(w =0 on Ω Introducing the new unknown y = b w L 2 (Ω and the operator Z : L 2 (Ω L 2 (Ω, Z(y = J(b y, the optimality system (511 is equivalent to the NCP (55; their solutions are related via the identity w = b y Now choose p such that (512 p (2, ] if n =1, p (2, if n =2, and p (2, 2n/(n 2 if n 3 Then the continuous embedding H 1 0 (Ω L p (Ω holds We have Z(y =G(y+λy, where G(y =(A 1 (A 1 (y b+u d +λ(w d b Note that G maps L 2 (Ω continuously affine linearly to H0 1(Ω + L (Ω L p (Ω (continuous embedding Next, consider a solution y of the NCP If y(x =0then 0 Z(y(x =G(y(x+ λy(x = G(y(x Ify(x 0then y(x > 0 and Z(y(x = 0, which implies y(x = λ 1 G(y(x > 0 This shows y = max{ λ 1 G(y, 0} L p (Ω Therefore, with p as in (512, the NCP corresponding to the control problem has the following properties: (a Any solution of the NCP lies in L p (Ω, with p>2as in (512 (b Z : L 2 (Ω L 2 (Ω is continuous affine linear (c Z(y =G(y+λy, where G : L 2 (Ω L p (Ω, p>2as in (a, is continuous affine linear In particular, Z maps L p (Ω continuously affine linearly to L p (Ω From these results we immediately can derive the assumptions (a, (b and (a in Example 55 In fact, from (a we see that we can pose the problem in L p (Ω instead of L 2 (Ω Now let q = p and r =2 Then (b shows that Z maps L p (Ω continuously affine linearly to L r (Ω, and thus condition (a of Example 55, and even condition (a with α =1hold From (c we conclude that Z maps L p (Ω continuously affine linearly to L q (Ω with q = p This establishes condition (b of Example 55 The control problem of the previous example is further considered in Section 62 REMARK 57 In Example 56 we saw that NCPs arising in practice sometimes satisfy stronger assumptions than those stated in Example 55 A typical situation is the following: The NCP is posed in the Hilbert space L 2 (Ω and Z : L 2 (Ω L 2 (Ω is continuously Fréchet differentiable Further, one can find p, q > 2 such that Z maps L p (Ω locally Lipschitz continuously to L q (Ω Finally, any solution of the NCP can be shown to lie in L p (Ω This is the situation we had in Example Proof of Theorem 52 We can simplify the analysis by exploiting the following fact

18 16 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES LEMMA 58 Let the Assumptions 31 hold and suppose that the operator Λ:u i Lq i (Ω ψ(u L r (Ω is semismooth at u = F (y Then the operator Ψ:Y L r (Ω defined in (18 is semismooth at y Further, if the Assumptions 34 hold and Λ is α-order semismooth at u = F (y then Ψ is α-order semismooth at y Proof We first observe that, given any M Ψ(y + s, there is M Λ Λ ( F (y + s such that M = M Λ F (y + s In fact, there exists a measurable selection d L (Ω m of ψ(ω such that M = i d i F i (y +s, and obviously M Λ : v i d iv i yields an element of Λ ( F (y + s with the desired property A more general chain rule will be established in Theorem 72 Setting u = F (y, v = F (y + s F (y, and w = F (y + s,wehave sup Ψ(y + s Ψ(y Ms L r M Ψ(y+s sup Λ(w Λ(u M Λ F (y + ss L r M Λ Λ(w sup Λ(w Λ(u M Λ v L r M Λ Λ(w + sup M Λ Λ(w M Λ ( F (y + s F (y F (y + ss L r By the local Lipschitz continuity of F and the semismoothness of Λ, we obtain ρ Λ = o( v Πi L q i =o( s Y as s 0 in Y def = ρ Λ + ρ MF Further, since d L ψ Bm L by Lemma 47, we have by Assumption 31 (a ρ MF L r L ψ F i(y + s F i (y F i (y + ss i L r L ψ c r,r i (Ω F i (y + s F i (y F i (y + ss i L r i = o( s Y as s 0 in Y This proves the first result Now let the Assumptions 34 hold and Λ be α-order semismooth at u = F (y Then ρ Λ and ρ MF are both of the order O ( s 1+α Y, which implies the second assertion For the proof of Theorem 52 we need, as a technical intermediate result, the Borel measurability of the function (513 ρ : R m R m R, ρ(x, h = max ψ(x + h ψ(x z T h z T ψ(x+h We prove this by showing that ρ is upper semicontinuous Readers familiar with this type of results might want to skip the proof of Lemma 59 Recall that a function f : R l R is upper semicontinuous at x if lim sup f(x f(x x x Equivalently, f is upper semicontinuous if and only if {x : f(x a} is closed for all a R LEMMA 59 Let f :(x, z R l R m R be upper semicontinuous Moreover, let the multifunction Γ:R l R m be upper semicontinuous and compact-valued Then the function g : R l R, g(x = max f(x, z, z Γ(x

19 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES 17 is well-defined and upper semicontinuous Proof Forx R l, let (z k Γ(x be such that lim f(x, z k= sup f(x, z k z Γ(x Since Γ(x is compact, we may assume that z k z (x Γ(x Now, by upper semicontinuity of f, f ( x, z (x lim sup f(x, z k = sup f(x, z f ( x, z (x k z Γ(x Thus, g is well-defined and there exists z : R l R m with g(x =f ( x, z (x We now prove the upper semicontinuity of g at x Let (x k R l tend to x in such a way that lim g(x k = lim sup g(x, k x x and set z k = z (x k Γ(x k By the upper semicontinuity of Γ there exists (ẑ k Γ(x with (ẑ k z k 0 as k Since Γ(x is compact, a subsequence can be selected such that the sequence (ẑ k, and thus (z k, converges to some ẑ Γ(x Now, using that f is upper semicontinuous and ẑ Γ(x, lim sup g(x = lim g(x k= lim f(x k,z k = lim sup f(x k,z k f(x, ẑ g(x x x k k k Therefore, g is upper semicontinuous at x LEMMA 510 Let ψ : R m R be locally Lipschitz continuous Then the function ρ defined in (513 is well-defined and upper semicontinuous Proof Since ψ is upper semicontinuous and compact-valued, the multifunction (x, h R m R m ψ(x + h is upper semicontinuous and compact-valued as well Further, the mapping (x, h, z ψ(x + h ψ(x z T h is continuous, and we may apply Lemma 59, which yields the assertion Proof of Theorem 52 By Lemma 58, it suffices to prove the semismoothness (of order β of the operator Λ:u i Lq i (Ω ψ(u L r (Ω (a Semismoothness: In Lemma 510 we showed that the function ρ : R m R m R, ρ(x, h = max ψ(x + h ψ(x z T h, z T ψ(x+h is upper semicontinuous and thus Borel measurable Hence, for u, v i Lr i (Ω, the function ρ(u, v is measurable We define the measurable function a = ρ(u, v v {v=0}

20 18 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES Since ρ ( u(ω,v(ω =0whenever v(ω =0, we obtain Furthermore, (514 a(ω = ρ ( u(ω,v(ω v(ω {v=0} (ω = Due to the Lipschitz continuity of ψ, wehave ρ(u, v =a v 1 (515 ρ(x, h 2L ψ h 1, ( o v(ω 1 0 as v(ω 0 v(ω {v=0} (ω which implies a 2L ψ BL Now let (v k tend to zero in the space i Lq i (Ω and set a k = a v=vk Then every subsequence of (v k contains itself a subsequence (v k such that v k 0 ae on Ω By (514, this implies a k 0 ae on Ω Since (a k is bounded in L (Ω, we conclude lim k a k L t =0 for all t [1, Hence, in L t (Ω, 1 t<, zero is an accumulation point of every subsequence of (a k This proves a k 0 in all spaces L t (Ω, 1 t< Since the sequence (v k, v k 0, was arbitrary, we thus have proven that, for all 1 t<, Now we can use Hölder s inequality to obtain a L t 0 as v Πi L q i 0 (516 ρ(u, v L r (Ω av i i L r a i L p i v i L q i ( max a L 1 i m p i v Πi L = o ( v qi Πi L qi as v Πi L q i 0, where p i = q ir q i r if q i < and p i = r if q i = Note that here we exploited the fact that r<q i This proves the semismoothness of Λ (b Semismoothness of order β: We now suppose that the Assumptions 34 and, in addition, (53 hold First, note that for fixed ε>0 the function (x, h R m R m ρ(x, h ε α h 1+α 1 is upper semicontinuous and that the multifunction x R m ε B m 1 is compact-valued and upper semicontinuous Hence, by Lemma 59, the function ( x R m max ρ(x, h ε α h 1+α 1 h 1 ε is upper semicontinuous and therefore Borel measurable This proves the measurability of the set Ω ε appearing in (53 For ε>0and 0 <β αwe define the set { Ω βε = ω : ρ ( u(ω,v(ω } >ε β v(ω 1+β 1, and observe that Ω βε Ω ε { v 1 >ε} def =Ω ε Ω ε

21 M ULBRICH: SEMISMOOTH NEWTON METHODS IN FUNCTION SPACES 19 In fact, let ω Ω βε be arbitrary The nontrivial case is v(ω 1 ε We then obtain for h = v(ω ρ ( u(ω,h >ε β h 1+β 1 = ε α ε α β h 1+β 1 ε α h α β 1 h 1+β 1 = ε α h 1+α 1, and thus, since h 1 ε, max h 1 ε ( ρ ( u(ω,h ε α h 1+α 1 showing that ω Ω ε In the case q 0 = min q i < we derive the estimate 1 i m > 0, µ(ω ε =µ({ v 1 >ε} ε 1 v q 0 1 L q 0 (Ω ε ( ε q 0 max c q0,q i (Ω ε q0 v q 0 i Π i L q i = ε q 0 O ( v q 0 Π i L q i If we choose ε = v λ Π i L q i, 0 <λ<1, then ( ( µ(ω βε µ(ω ε +µ(ω ε=o v γλ Π i L q i + O v (1 λq 0 Π i L q i This estimate is also true in the case q 0 = since then µ(ω ε =0as soon as v Π i L q i < 1 This can be seen by noting that then for aa ω Ω holds v(ω 1 v 1 L v Πi L q i v λ Π i L q i = ε Introducing ν = q 0 r q 0 r if q 0 < and ν =1/r, otherwise, for all 0 <β α, we obtain, using (515 and Lemma A1 ρ(u, v Lr (Ω βε 2L ψ v 1 Lr (Ω βε 2L ψc r,q0 (Ω βε v L q 0 (Ωβε m (517 2L ψ µ(ω βε ν v L q 0 (Ωβε ( ( m = O v 1+γλν Π i L q i + O v 1+(1 λνq 0 Π i L q i Again, we have used here the fact that r<q 0 q i, which allowed us to take advantage of the smallness of the set Ω βε Finally, on Ω c βε, (1 + βr q 0, 0 <β α, holds with our choice ε = v λ Π i L q i ρ(u, v L r (Ω c βε ε β v 1+β 1 ( = O v 1+β(1 λ Π i L q i c q r, 0 (Ωc Lr (Ω c βε 1+β βε v βλ Π i L q i v 1+β L q 0 (Ω c βε m Therefore, ρ(u, v L r ( = O v 1+γλν Π i L q i ( + O v 1+(1 λνq 0 Π i L q i ( + O v 1+β(1 λ Π i L q i We now choose 0 <λ<1and β>0with β α, (1 + βr q 0 in such a way that the order of the right hand side is maximized In the case (1 + αr q 0 the minimum of all three exponents is maximized for the choice β = q 0 r = νq r 0 and λ = q 0 γ+q 0 Then all three exponents are equal to 1+ γνq 0 γ+q 0 and thus ( (518 ρ(u, v L r = O v 1+ γνq 0 γ+q 0 Π i L q i