Perturbed optimization on product spaces

Voohnwr

Ina,v,r.>, Theorv. A4rrhod.i & Apphcurrons,

0362-546X(94)0021

PERTURBED

(Recerved

9 August

01 Mathematics, lYY3; received

Key words and phrusrs: accretive operators

l-8

OPTIMIZATION

MACIEJ Department

KOCAN University

in revised

Viscosity

form

solutions,

ON PRODUCT

and ANDRZEJ of California,

Barbara,

1993; received lemma,

SPACES-f

SW@CH$

Santa

1 December Ekeland

Vol. 26, No. I, PP. 81-90. 1996 Elrer~r Science Ltd Printed in Great Britain

CA for

optimization,

93106,

publication nonlinear

U.S.A. 18 August

1994)

semigroups,

0. INTRODL~‘TlON

The motivation for the optimization result presented in this paper comes from the theory of viscosity solutions of partial differential equations in infinite dimensions; see the Introduction in [l] for remarks about typical equations which are investigated by the viscosity method. The definition of viscosity solutions involves minima and maxima of certain functions and the proofs of uniqueness depend on the ability to produce extrema of special types. Due to the lack of local compactness in infinite dimensions, continuous functions in general do not assume their extremal values and there arises a need for perturbed optimization: given a closed subset D of a Hilbert space and a bounded continuous function f: D + IR, find an appropriate small perturbation p: D + IR such that f + p attains its minimum over D. As explained in [I], when dealing with “bounded” problems, the most useful optimization technique is that due to Ekeland-Lebourg-Stegall: p is a linear functional of small norm. Recently, another result in this direction was obtained by Deville et al., see [2]. The “unbounded” case requires a different approach. In the simplest such case, the equation to be solved contains a linear, densely defined, maximal monotone operator A and the perturbation p has to behave nicely on the trajectories of the semigroup generated by -A. An appropriate optimization technique for this case was introduced by Tataru in his basic paper [3]. Let us explain this result, see [4] for information about nonlinear semigroups. Suppose that H is a real Hilbert space and let A be a maximal monotone (equivalently, m-accretive) operator in H. Then -A generates a strongly continuous semigroup S(t) of contractions on D(A), where D(A) denotes the domain of A. For (x, y) E H x D(A) define

d is almost a metric (it lacks symmetry). The following proved by Tataru (see [l, 31). 1. If u E LSC(D(A)) such that the map

LEMMA

analog of Ekeland lemma [5] has been

is bounded from below, then for every E > 0 there exists 2 E D(A) x - u(x)

t

&d(X, X)

t This work \cas supported in part by Army Research Office Contract DAAL03-90-G-0102 and National Science Foundation Grant DMS90-02331. We would like to thank Professor M. G. Crandall for his support and good advice. $ Current address: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, U.S.A. Xl

82

has a minimum

M. KOCAN

and A. SWIECH

over D(A) at ,? and u(i) < inf u + E. L’(A)

The Tataru distance d is Lipschitz but not differentiable and, therefore, is not suitable for working with second order equations. The perturbation technique which we are about to present is a combination of Tataru’s and Stegall’s and can be used to generate perturbed minima of a function defined on a product space. Its virtue is the property that our perturbations p are “half-smooth” and can be used to investigate equations in separated form, where the unbounded term and the second order term involve different variables. This issue will be addressed in the future. We think however that the perturbation result is interesting on its own. Note that if A = 0 then d(x,y) = [Ix ~ y/l f or all x, y E H and in this case we obtain a combination of Ekeland’s and Ekeland-Lebourg-Stegall’s results. So let K be another real Hilbert space. We will prove the following theorem.

THEOREM. Suppose that D c D(A) x K is closed and the projection of D onto K is bounded, that is there exists f. > 0 such that if (x, y) E D

then I1yIl I L.

Let F: D -+ IR U {al be lower semicontinuous there exist (2, j) E D and p E K satisfying

(1)

and bounded from below. Then for every E > 0

IIPIi 5 E

(2)

such that the map (X, Y) ++F(X, Y) + Ed&, i) + (P, Y> has a strict global minimum

over D at (j?,j) and F(i,j)

< i;f F + E.

I. EKELAND-[LEBOURG

(3)

LEMMA

The proof of our theorem requires the following lemma, which can be easily extracted from the fundamental paper of Ekeland and Lebourg [6]. We include the proof for the reader’s convenience.

LEMMA 2. Suppose that T C D(A) x K is closed and the projection of Tonto K is bounded, that is there exists f. > 0 such that if (x,y) E T Let Y: T + R U 1~01be lower semicontinuous

then 1Iy)l I L. and bounded from below. For p E K define

F(P) = @‘l:f, Tw(x.Y)

+ (P,Y)l.

(4)

Perturbed

83

optimization

For E > 0 let G, = (p E K: 3 q > 0 such that (xiryr) E T, Y(x,,y;)

+ (p,yi)

< F(p) + V,

i = 1,2, implies llyr - yzll I El. Then G, is dense in K. Proof. First we will recall the definition of local c-supportability from [6]. We will say that F is locally a-supported at 5 E K if there exist 6 > 0 and p* E K such that F(P) 1 F(ti) + (P*,P

- j> - ~11~- bii

for IIp - j?II 5 6.

It has been shown in [6] that if F is lower semicontinuous then the set S, of is locally e-supported is dense in K. To prove this statement first one uses to localize things and then one define a differentiable “barrier function” Ekeland lemma. (4) implies that our function F is Lipschitz continuous (with Lipschitz therefore, S, is dense. We will prove the lemma by showing that

(5)

all points where F the norm on K to uses the standard constant L) and,

G,, 1 S,.

(6)

To this end suppose that i, E S, and let p* E K and 6 > 0 be such that (5) holds. Let q = aa/2 and suppose that (x, y) E T satisfies ‘W,Y)

+ (5, Y> < F(b) + rl.

(7)

We will show that then (8) which implies that the set of the y-components of all q-optimal diameter at most 3c, which is equivalent to j? E G,, . Let us take IIp - @]I 5 6. Then from (7)

with respect to F(p) (x, y)s has

F(P) - F(b) 5 ‘Kc y) + (P, Y> - F(b) < ‘uk

Y) + (P, y> - ‘ye, Y) - + rl

= (P - b,Y)

+ rl.

On the other hand, by (5) we have F(P) - F(6) 2 (P*, P - I% - ~11~ - till. Hence for all IIp - FII 5 6 (P* - Y,P - !2 < EIIP - 511 + ;, which implies (8) and, consequently,

(6). The proof of lemma 2 is complete.

n

84

M.

KOC‘AN

and

A. SWlECH

Remark. It is not difficult to see that G,s are open in K. Suppose, using notation that T C K and Y is a function of y alone. Define

of lemma 2,

Then G is a dense Gb (hence second category) subset of K and if p E G then all minimizing sequences for F(p) are Cauchy, therefore, the map T3y

- WY) + (P,Y>

has a strict global minimum over T. In [6] this result is obtained as a corollary from a more general theorem, the proof of which adapted to this simple case follows the outline described above. More general version, with Tpossessing the Radon-Nikodym property, is due to Stegall 171. 2.

PROOF

OF

THF

THEOREM

Now we will proceed with the proof of the theorem. This proof uses the ideas borrowed from the proof of Borwein-Preiss theorem [8], as presented by Li and Shi, see [9]. Proof of theorem. First of all choose a sequence lsi]:=,

of positive real numbers such that

Choose (x, , y,) E D such that F(x,,y,) Let TO = D. We will inductively subsets of D. Fern= 1,2,...,oolet

< infF+ D

i.

(10)

construct sequences (x,, yi) E D, pi E K and a sequence T of

= F(X9Y) + i djd(X, Xl) + f (Pi7Y I =I i= I

yn(X,Y)

-

Yi>.

Note that for n = 1, 2,. . .

(11)

~rllnX,~Y,) = ul,-,(X,.Y,). Consider the map T, 3 (x, Y) - F(x, y) + 6,4x,

x,).

From lemma 2, applied to T = 7;,, Y(x, y) = F(x,y) + S,&, a/2 A &6,/4L A &/16L such that there exists an ‘I, > 0 such that diam J: 3 X such that (3, p) E T,, and F(X, j) + a,@, <

inf (X,Y)

t

IF(x,y) 70

+ 6,d(x,x-,)

+ (p,,y

xi),

we can find

x,) + (p,, j - y,)

- y,)) + VI

5 +. 1

jIplII 5

Perturbed

85

optimization

Define T, =

Choose (x,,y&

(x,y)

E To: Y,(x,y)

5 inf Y, + VI TO 1

E T, such that

Given L1, (x,,Y,), &,Y,), . . . . (x~~~,~~~,),P,,P~ ,..., pnml,(xn,yn)~ use lemma 2 to construct pn E K and qn > 0 such that

G-,,

firstwewill

and diam J: 3 X such that (X, j) E T,-, and Y,,(X, j) <

inf Ynu,(X,Y) + 1;1, W.-V) E Tn-, 1

(14)

Let

Note that

and choose (16) such that (17) Continue

this process. By construction

we have for all n that

The sets T, are closed (because F( +, *) and d(. , .) are lower semicontinuous), nested: T,,+, E Tn. Suppose that (.V,J), (X, ,F) E Tn. (14) yields

Thus the sequence (y,l is Cauchy and, therefore,

nonempty

and

M.

86

KOCAN

and

A. SWlECH

Now (X, .V) E T, , (15) and (16) imply that V(%

x/J + ‘u,- I(-%J) + (p,, P - Y,> = ul,(X,V) 2 pf y?l + VII 5 nI

Therefore,

since (X, J) E T, E T,_, and inf, ~,G,%J

5 Yn-l(X,,Yn)

+ IJn =

‘fl,(%l,Y,)

, Y,-,

= infrnez Y,-l,

- y,-,(%J)

+ vn.

y,-,(X,,Y,)

we have

+ (P,>Y, -R

+ v,

This shows that for every (X, J) E Tn.

(21)

In particular

which implies (see [I]) that there exists X-E D(A) such that x,, + i.

(22)

Since T,s are closed, from (20) and (22) we obtain that (X,j) E fir= I T, in fact

C

D. We will see that

,,(-Jr,, - IGlAl.

(23)

We saw in (19) that the diameters of the projections of T,s onto K converge to 0. Thus to justify (23), by (21) it is enough to show that 4% x,) + 0,

d(i, x,) ---t 0

implies X = 2.

This will follow from the fact, that the semigroup S(l) is contractive. sequences t, 1 0, s, 1 0 such that lb - W,)x,II

* 0.

(24) Indeed, there exist

(‘x - S(s,)x,l( --f 0

as n --t 00. However, then ~1S(s,)X~ S(t,,)h.// 5 lIs(t, + .SJX, - S(~,,Pll + IIW, + %)Xn - W,)~ll % (/S(f,)X,~ --- iq + I(S(s,)x, - 2)) --t 0. Now observe that S(s,,).V+ X and S(t,,),? * X and (24) follows. We will see that the pair (E, j) has the desired properties. First we will show that (j?,j) is the strict minimum of ‘Pmover D. So suppose that (X, J) # (i, j). Then by (23) there exists n,, L 1

Perturbed

87

optimization

such that (X, j) E T,,,- ,‘I~,~~, which implies, by (15), that Y&K J) > jnf Yn,, + II,,, . ‘1,)’ However,

from (17) we have

and, therefore,

and, consequently,

due to (18), for all q > n,,

We will let q + 03. It is not difficult

to see that

because Also

Since F is lower semicontinuous,

we obtain

lim inf YV(xy, .vy) = lim inf F(q,, -Y,) + i 6,&, q-a Y-a I= I 1

F(X,j)

+

E d,d(i,x,) I- I

+

i I-

, X,) + i i= (P,,j

(Pi,Y,

- Yi)

I

-Y,)

I

(25) We conclude that Yn,,(,i;,j) 2 Y&z, p, + 2

.

Therefore,

= Yno(X,J) +

1 d;d(X, x,) + C (Pl7Y - Yi) I =PI,) +I ,=“()-I

2 Y&i-,)

-g (P,,.V -Y,> I = n,)+I

+

+ 2

88 However,

M. KOCAN

and A. SWIECH

by (12) and (13),

which implies

This shows that (.C,j) is a minimum of ul, over D. This minimum were a minimizing sequence for Y, over D such that

is strict, because if (xk, y,J

then one could find n, as above which worked for all ks on a subsequence and then argue as above to reach a contradiction. Define

Then (12) yields (26)

which shows (2). Now drop Sy=, (p,,y,)

in U; to conclude that the map

(XTY) - F(x, .Y) + i

6id(X, Xi) + (py y)

has a strict minimum over D at (.C-,j). From the triangle inequality for d(., .) and (9) we see that

$ d,d(X,.Y,) -- i 6,d(i,x,) I: I ,= I

= i 6,(m, I= I

I

i 6,d(x,i) ,= I

Xi) - 4% Xi))

Perturbed

89

optunization

This completes the proof of the first part of the theorem. The second part will follow from this chain of inequalities, which uses (lo), (18), (25), (12) and (1)

=F(i

3,G)+(p,j)-c

8

Therefore, F(i, j) + (p, .V) < i;f F + f + 5 and from (26)


t I:

and the proof of (3) and of the theorem is complete.

n

Remark. To keep the exposition clear we stated our theorem in the proof remains valid (with trivial modifications, e.g. p E K*, not p Banach and K* has a differentiable “bump function”, see [6] for However, from the point of view of the theory of viscosity solutions Hilbert space case is the most interesting.

Hilbert space setting. The E K), if H and K are just definition and examples. in infinite dimensions, the

Another remark is that if A = 0 then d(x-, .v) = /IX ~ yil for all x, y E H, and in this case one can prove analogue of our theorem by a category argument using ideas from 121.

1. CRANDALL M. G. & LIONS P.-L., Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. VI. Nonlinear A and Tataru’s method refined (to appear). 2. DEVILLE R., GODEFROY G. & ZIZLER V., A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. .funcl. Analysis 111, 197-212 (1993). 3. TATARU D., Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms, J. math. Analysis Applic. 163, 345-392 (1992). 4. BARBU V., Nonfinear Semigroups and Differenirul Equations in Bonach Spaces. Nordhoff, Leyden (1976). 5. EKELAND I., Nonconvex minimization problem,. Buff. Am. mafh. Sot. 1, 443-474 (1979).

90

M. KOCAN

and A. SWIECH

6. EKELAND I. & LEBOURG G., Generic Frechet differentiability and perturbed optimization problems in Banach spaces, Trans. Am. math. Sot. 224, 193-216 (1976). 7. STEGALL C., Optimization of functions on certain subsets of Banach spaces, Math. Annln 236, 171-176 (1978). 8. BORWEIN J. & PREISS D., A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Am. math. Sot. 303, 517-527 (1987). 9. LI Y. & SHI S., A generalization of Ekeland’s c-variational principle and of its Borwein-Preiss’ smooth version (to appear).