Aubin, J.-P.

International Institute for Applied Systems Analysis (IIASA)

Variational Inequalities RevisitedAubin, J.-P.IIASA Working PaperWP-83-043April 1983 Aubin, J.-P. (1983) Variational Inequalities Revisited. IIASA Working Paper. WP-83-043 Copyright © 1983 by the author(s). http://pure.iiasa.ac.at/2269/ Working Papers on work of the International Institute for Applied Systems Analysis receive only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage. All copies must bear this notice and the full citation on the first page. For other purposes, to republish, to post on servers or to redistribute to lists, permission must be sought by contacting repository@iiasa.ac.at NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR VARIATIONAL INEQUALITIES REVISITED Jean-Pierre Aubin April 1983 WP-83-43 Working Papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily repre- sent those of the Institute or of its National Member Organizations. INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria VARIATIONAL INEQUALITIES REVISITED Jean-Pierre Aubin Let K be a closed convex subset of a reflexive Banach spaceX * and A be a set-valued map from K to X satisfying ( A is finitely upper-semicontinuous with nonempty closed convex bounded images. Our purpose is to solve variational inequalities (or generalized equations) - i) x E K  ii) OEA(X) + N ~ ( ~ T )  * where N~(x) := {PEX /sup (p.x-y) - > 01 is the normal cone to K Y= at x E K ,  by balancing a) the lack of boundedness of K, measured by its "barrier cone" b(K), defined by (because the larger b(K), the lesser is K unbounded) b) with t h e  d e g r e e  of monotonicity of A, measured by a nonnegative proper lower semicontinuous function B from X to IR U {+a) satisfying We shall say that such a set-valued map A is B-monotone. We * ( 2 )  denote by B its conjugate function . For instance, we can take * * [ i) B(z) := 0 (and thus, B = ${Oj ,Dam B = {o)) ( 3  * * ii) B(z) := l l  zll (and thus, B = ylgt,Dorn B = B,) (4 1 * 1 a* 1 1 * * I I ~ I I ~  (and thus, B = - I I - I I  ,-+-=1,DomB =X ) (iii) B(z) := - aa* a a* In the following theorems, we shall measure the degree of mono- * tonicity of G through the size of the domain of B : the larger * Dom P , the more "monotone" is G. T h e o r e m  I .  We posit assumptions (1). Assume that A is B- monotone.and that (6 0 E Int (b(K) + A(K) + Dom B*) Then there exists a solution EK to the variational inequality 0 E A (3 + NK ( j t )  . A Assumption (6) shows how the lack of boundedness of K is compensated by the degree of monotonicity of A. We point out that (6) is satisfied when one of the following instances is satisfied. * i) K is bounded (b(K) = X ) * ii) A is surjective (A(K) = X ) ( 7 )  C iii) A satisfies (4) with ~ ( z )  := - a llzlla, c > 0, a > 1 iv) A satisfies (4) with B (z) := cl zll , c > 0 and Naturally, these examples are known (See Brgzis (19681, Lions (1969) and Browder (1976)). The novelty lies in the introduction of the function 6 as a parameter in assumption (6). We recall that NK(x), the normal cone to K at x, is the subdifferential ( 3 )  of the indicator $ J ~ .  Therefore, variational inequalities are particular cases of inclusions of the form when V :x+IRU{+W) is a proper lower semicontinuous convex func- * tion and A is a set-valued map from the Banach space X to X , which were studied by Brgzis-~araux (1976) , when A is maximal monotone, for solving Hammerstein equations (see Brgzis-Browder (1976)!. We shall extend Theorem 1 to this case. To this end, we assume once and for all that (9 Dom V C Dom A and we observe that a necessary condition for the existence of a solution x to (8) is that We shall prove that this condition is "almost sufficient". Theorem 2. We posit assumptions (1). Assume moreover that A is 6-monotone. Then there exists a solution x to the inclusion ( 8 when * * (11) f E Int (Dom V + A Dom V + Dom 6 ) . * Remark. The size of Dorn 6 balances the interiority condition in assumption (111, as the following corollary shows. Coro ZZary 3. We posit assumptions (1 ) . a) ~f A is monotone, (i.e., B=O), then * (12) * Int (Dom V + A Dorn V) C Im (A + aV) CDom V + A Dorn V . b) If there exists c > 0 such that then * (1 4 )  ~m (A+ av) =  DO^ v + A  DO^ v c) If there exist c > 0 and a > 1 such that (15) c a V(x,p) , (y,q) €graph (A), (p-q,x-y) - > - a llx-yll then * * (16) ~m (A + av) =  DO^ v + A  DO^ v = x I Before proving Theorem 2, we shall characterize problem (8) by equivalent problems. For that purpose, we associate to the function V, to the map A and to an element f E X  the function @ defined on Dorn V by: We observe that * since, for all UEA(Y), V(y) + V (f-u) - (f-u,y) > 0, thanks to - the Fenchel inequality. We can also characterize the set-valued map A by the function y defined on Dorn AxDorn A by Proposition 4. Assume that the images A(x) are nonempty, closed, convex and bounded for all xEDom V. The following problems are equi- valent i) -I ZE~orn v such that ~ E A Z  + av(Z) * ,  ii) El p E Dom V such that f E 5 + AaV (p) (20) iii) ri Z E  om V such that Vy E Dom V, Y (Z,Y) - <f,Z-y) + v(Z) -V(y) - < 0 iv) =I x E Dom V such that @ (x) = 0 (= min @ (y) )yEDom V Proof. a) Let be a solution to (20) i) : then there exists - p E aV (x) such that f -p E AZ c A ~ V *  (p) . Conversely, let p be a * - solution to (20) ii) . Then there exists ZE aV (p) such that f E p + Ax. Since p E 3~ (Z) , then f belongs to av (Z) + Ax. b) Let Z be a solution to (20) i) . There exists EE A(Z) such that f€aV(x) + u, i.e., such that VyEDom V, By taking the infimum on ~ ( x )  , we deduce inequalities (20) iii) . c) Inequality (20) iii) can be written inf inf [V(X) - ~ ( y )  - ( f-u,x-y )I < 0 . - yEDom V uEA (Z) Since Dom V is convex, A(Z) is convex weakly compact, the lop- sided minimax theorem implies that SUP sup [v(Z) -V(y) -(f-u,Z-y)l uUll(x) yEDomV * = inf [v(Z)+V (f-u) -(f-u,x)] = @(x) uEA (x) Hence @ (x) < 0. - d) Let ZE   om V satisfy @ (x) = 0. Since A (x) is weakly compact and V is weakly lower semicontinuous, there exists - u E A (x) such that This is equivalent to saying that: f-EE a~(x), i.e. that x solves (20) i) . The equivalence between (20) i) and (20) iv) allows to inter- pret the solutions to problem (8) as a solution to a minimization problem (minimization of the functional 4) and provides a vari- ational principle. The equivalence between (20) i) and (20) iii) allows to solve problem (8), (and, in particular, variational inequalities) by applying minimax inequalities to the function defined by We observe that i) Yx, y -+ 4 (x,y) is concave (22) ii) Yyt4(y,y)=0 that @ is "monotone" in the sense that and that (24) ~y E X, .x -+ 4 (x, y) is lower semicontinuous (1 for the finite topology . Therefore, if Dom V were compact, we could apply the generalization of the Ky Fan inequality (1972) due to ~rgzis-~irenberg-~tampacchia (1972), which would imply the existence of a solution ZE~orn Vto the inequalities (20) iii) , i.e., a solution x to problem (8) . When Dom V is not compact, we shall prove by approximation that assumption (11) is sufficient for the existence of a solution to inequalities (20) iii) . Proof of Theorem 2. We set Kn := {x E Dom V/V (x) - < n and ll xll - < n}. The subsets Kn al are weakly compact and convex and Dom V = U Kn because X is n=l reflexive. Since Kn is weakly compact and convex, Ky  an's inequality for monotone functions implies that, for all n - > 1, there exists xn E Kn solution to thanks to properties (22), (23) and (24). We shall now use assumption (11) for proving that xn remains in a weakly compact subset of X. For that purpose, thanks to the uniform boundedness theorem, it is sufficient to prove that * (26) V ~ E X  , lin(p) such that sup (p,xn) < +m . n - > n (p) * * By assumption (11), there exist q > O f  rEDom B , qEDom V , yEDom V, uEA(y) such that We choose n(p) to be the smallest n such that yEKn. By taking the duality product with xn we get * We use ~enchel's inequalities (r,xn -y) 5 B(xn -y) + B (r) and * (q,xn) 5 V(xn) + V (9). We obtain S i n c e  A i s  6-monotone, w e  deduce t h a t  y ( x n , y )  - ( u I x n - y ) =  i n f  ( p - u , x n - y ) l B ( x n - y )  . pEA (x,) T h e r e f o r e ,  i n e q u a l i t y  (29)  becomes Consequent ly ,  f o r  a l l  n  - > n ( p )  , w e  deduce from (25)  t h a t  * * The r i g h t - h a n d  s i d e  i s  f i n i t e  because  rEDom f3 , qEDom V and yEDom V. Hence t h e  sequence  i s  bounded and t h u s ,  weakly r e l a -  t i v e l y  compact. So, a  subsequence  o f  e l e m e n t s  xn ,  converges  weakly t o  some - x E X .  S i n c e  V is  lower  semicon t inuous ,  w e  deduce from t h e  mono- t o n i c i t y  o f  A and from t h e  v a r i a t i o n a l  i n e q u a l i t i e s  t h a t  V ( Z )  - < l i m i n f  V(xn)  n  < l imsup  [V(y)  + ( f , x , - ~ )  + Y ( Y . X , ) I  - n - T h e r e f o r e ,  x  be longs  t o  Dom V and (31 VyE Dom V ,  0 - < 4 (y ,Z)  . d )  W e  deduce from p r o p e r t i e s  (22)  and (23)  t h a t  Indeed,  i f  t h e  co n c lu s ion  i s  f a l s e ,  t h e r e  would e x i s t  z  E  Dom V such t h a t  0  < @ (x, z )  and by (24)  t h e r e  would e x i s t  - t € ] O , 1  [ such t h a t  - By t a k i n g  y  = x + z ( z - x ) ,  i n e q u a l i t y  (31)  i m p l i e s  t h a t  Hence, t h e  c o n c a v i t y  of  @ w i t h  r e s p e c t  t o  t h e  second v a r i a b l e  y i e l d s  t h a t  a  c o n t r a d i c t i o n  t o  ( 2 2 ) i i ) .  Then P r o p o s i t i o n  4 i m p l i e s  t h a t  t h e  s o l u t i o n  x of  (32)  i s  a  s o l u t i o n  t o  t h e  problem ( 8 )  . I Notes (1) The finite topology on a convex subset N of a vector space is the topology for which the maps BK from the simplex n sn := {AEIR~ I Ai=l} to N defined by i= 1 are continuous for all finite subsets K := {xl, ..., xn} of N. It is stronger than any vector space topology and any affine map is continuous for the finite topology (see Aubin (1979), * 7 . 1 3 )  A finitely upper semicontinuous map from K to X is a set-valued map upper semicontinuous from K supplied * with the finite topology to X supplied with the weak *- topology. When A is finitely upper semicontinuous, then the map x -+ in£ (u,x-y) is lower semicontinuous on K for the uEA (x) finite topology (see Aubin (1979), 513.2.4). * (2) The conjugate function B of a function B:X+IR u { + a )  is * defined on X by * B (p) := sup [(p,x ) - B(x) I . XEX A function B is convex and lower semicontinuous if and only * if B = B . It satisfies the Fenchel inequality (3) The indicator of a subset K is the function $K defined by $K(~) = 0 when x E K and iK (x) = +a if not. (4) B, denotes the unit ball of the dual. ( 5 )  The subdifferential of a convex function V is the subset * of gradients of the affine functions x -+(p,x) - V (p) below V and passing through (x,V(x)). When V is ~gteaux- differentiable at x, then aV (x) = {VV (x) 1 .  The set of points x E X  for which aV(x) f a is dense in Dom V. References Attouch, H. Aubin, J.-P. 1979. Mathematical Methods of Game and Economic Theory. North-Holland. Bre'zis, H. 1968. Equations et ine'quations non line'aires dans les espaces vectoriels en dualite'. Ann. Inst. Fourier 18, 115-175. ~rgzis, H. E Browder, F. 1975. Nonlinear integral equations and systems of Hammerstein type. Adv. in Math. 18, 115-147. Br6zis, H. E Haraux, A. 1976. Image d'une somme d'opgrateurs monotones et applications. Israel J. Math. 23, 165-186. ~rgzis, H., Nirenberg, L. & Stampacchia, G. 1972. A remark on Ky Fan's minimax principle. Boll.Un.Mat. Ital. 6, 293-300. Browder, F. 1976. Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc.Symp.Pure Math. Vol. 18(2). Fan Ky 1972. A minimax inequality and applications. In Inequalities 3, Academic Press, New York, 103-113. Lions, J.-L. 1969. Quelques mgthodes de r6solution de problsmes non lingaires. Dunod, Paris. 

Variational Inequalities Revisited

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