EXISTENCE AND INF-COMPACTNESS

We now study the properties of F , the essential objective of P, and in particular we analyze the implications of Grinold's growth condition, Assumption 3.2 (or 3.3). W e first show that all trajectories

x

= (xt)E1 of interest for P a r e bounded in a certain normed space and then show that restricted t o that space the function F is weakly inf-compact from which the existence of optimal solutions follows immediately.

Note that if for all

t.

f t = f and the constraints implied

x c K , t = I

_t ,...,

or if we added a constraint of that type, with K C Rn compact and f bounded on K X K , then existence and related results could be obtained via the

standard method of successive approximations which also gives good error estimates [13, Chapter 61, [14. Chapter 41. In this paper we do not introduce such artificial (uniform) boundedness conditions on the trajectories (xt)E1. A fortiori, we shall not require that optimization takes place in the space 2: of bounded sequences in Rn. The appropriate space turns out t o be

a s confirmed by the results below; here

I 1

denotes the II 1 -norm in Rn, i.e.

The arguments rely on the asymptotic behavior of "averaged" trajectories.

Fix any 51 c (0. a

1.

Now t o each x = (x ^)0° we associate t t = l

and

Note that u and vT are convex combinations of (x

T 0,

.... .

^, xT) and (x

1' """

xT) respectively. and that y T and zT are just scaled versions of these vectors. We have that

while

Now observe that

Also

which means by (5.6) that

I f

1 1

x

I( = +

^a, the case which w i l l be of interest, then the

BT

converge monotonically t o

+

^00. This means that

-

¹

l i m

Ifl =

^q

T*

the convergence being from below. Also, and this only depends on having

BT

>

0 for T sufficiently large, every cluster point of the sequence {

1

yT

I,

T=l,

...

⁾ belongs to [0, 11. This means that

and

where B is the unit ball i n Rn, and hence each one of these sequences admits cluster points.

5.1 LEMMA. Su~pose P is proper and satisfies Crinold's qrowth condition assumptions 3.1 3.2), x = (xt); is such that I J x ( l =

+ -

and either

l i m sup lxt _t-ra,

1

^lit

< -

or there exists II c (0, a] such that

l i m z T = z T-ra,

exists with the zT as defined above (5.5). Then F(x) =

+-.

PROOF. The argument follows the same pattern as the proofs of [3, Theorem 4.1.1, [15, Theorem 11. We begin by showing that w i t h (5.12) and (5.1 3), the sequence { (yTelS zT), T=l.

...

^} admits a cluster point (y, z) with y = q'z, z+O and q' c [0, a]. Suppose first that (5.12) holds. Using (5.5), (5.9) and (5.8) we see that

F m m (5.10) we know that some subsequence of the sequence { zT. T=l.

...

^}

converges t o some z w i t h lzl

=

q'l. Since 8;' goes t o 0. it would follow that

lirn yT-l = q z

see [15, Lemma 11 for the details. Now suppose that (5.13) is satisfied. Then some subsequence of { IyT-l

I

T=l,

...

⁾ w i l l converge t o a 0 r: [O, 11.

For the rest of the proof we assume that actually

lirn ( Y ~ - ~ . z T = ('I'Z. Z) T-

with q'r: [0, a]; there is no loss of generality i n doing so since a l l assertions remain valid i f we work only with a converging subsequence. For the sake o f the argument, l e t us assume that

l i m sup _T-

E

q t-l f t ( ~ t - l , xt ) ( Y ( +

-

Since for all t, ft

2

h and h = hl, as defined by (3.2), is convex, from (5.2) we obtain

for T sufficiently large. Reexpressing this i n terms of (yT

-

xT) and dividing T -1

both sides by AT = (1 -11) (1-11 )

BT

yields

Since

BT

and AT tend t o

+-

with T, from (3.8) and the limiting properties of the sequence (yT-l. ~ ~ ) y - ~

-

we obtain

which contradicts Grinold's growth condition (3.7). Hence

l i m sup

c

:=1 qt-l f t ( x t-l , xt ) =

+- .

T-

But now recall that T-& a. therefore we also have

l i m sup T t-1

a ^{f t} ( ~ t - ~ . Xt) = F ( X I = +-.

T-

(the positive part of the sum already dominates the negative part with the parameter 11 less than a), and this completes the proof. a

The conditions (5.12) and (5.13) cover all cases that seem t o be of interest.

However, i t is possible t o generate trajectories, with more than exponential growth and for which the "averaged" trajectories { zT, T.1,

...

^{} do not}

converge. Such trajectories must have very wild tails! And for these,

Grinold's growth condition would not be sufficient t o guarantee that they correspond t o F(x) = ^+oo. we would need to impose much more constringent growth conditions t o handle such exotic cases.

By Lemma 5.1 we may safely restrict optimization to those sequences

00 1'

( X t ) t = l i n Pn (a) whose natural pairing is with 2.: This plays a significant role i n the convergence results of the next section but it also has some bearing on the question of the existence of optimal solutions. Weak convergence i n %,(a) 1 is characterized by having "componentwise"

U 00 sequence. This set-up w i l l provide us with the topological framework for the study of the properties of P. Henceforth, we w i l l think of P as being defined on Pn (a) with the essential objective function now given by: 1

00 otherwise

5.2 PROPOSITION. Suppose P is proper (Assumption 3.1). Then F is a proper, weakly lower semicontinuous function.

PROOF. Properness of F on Pn 1 (a) has been argued i n Section 3 in connection with Assumption 3.1. Lower semicontinuity is obtained as a consequence of a version of Fatou's Lemma. By Assumption 3.1. i, the function h ( i f for all t) is proper and convex, so let a be an affine function mqjorized by h. Then for all t,

Now consider a collection (xu c Pn 1 (a), u c N) converging weakly to x. For all u r N and T, we set

Since the quantities involved are nonnegative, we have that for a l l u, the ruT are monotonically nondecreasing with T and thus

1: :

=

l i m r

u (5.16)

u,T T-

is well defined, possibly with value ^too. Hence for all T

l i m inf l i m inf r , u r N r r N

then taking l i m sup with respect to T on both sides (which of course does not affect the right-hand side) and using (5.16), we obtain

lirn sup lirn inf r V m T

i

lim inf lirn sup r

Since A is an affine functional and the xv converge weakly, this inequality gives us the weak lower semicontinuity of F since i t implies that

lirn inf F(xU)

1

F(x).

a

vcN

The existence of optimal solutions could now very well be settled by requiring that dom F, the effective domain is contained in

a

weakly compact

set;

the level

sets

of F being weakly closed i t wou1.d give us t h e weak inf-compactness of F.

The following growth condition would suffice: for some sequence of nonnegative numbers At,

t=

1,

... ^with

we have for all x c dom F that lxtl ( At for all t. Then every trajectory i n dom F is uniformly summable and weak compactness then follows. see (16, p. 201 for example. In particular this argument shows that we may allow feasible

trajectories to grow exponentially at a rate less than a-l. When this is translated in the language of capital theory, the condition

Ix

I

(

\

with l i m sup (A3 l i t ( a-l t-

is nothing else than the requirement that the rate of impatience, as defined by Fisher (171 exceeds the rate of capital growth. Otherwise we would embark on a path of capital accumulation (31. We shall see however, that we do not need t o introduce weak compactness artificially. in fact it is already there i n some way, as a consequence of the assumptions we have been working with so far. We begin

If P is proper. we have that H is proper and weakly lower semicontinuous. as follows from Proposition 5.2 (replacing in the definition of F every f t with h).

Moreover, H is convex (convexity of h) and H(0) is finite.

5.3 LEMMA. Suppose P is proper and satisfies the strict version of Grinold's qmwth condition (Assumptions 3.1 and 3.3). Then H is weakly if-compact, i.e. for a l l

13cR

is weakly compact.

PROOF. Of course it suffices to consider the case when lev F is nonempty, since

P

H(0) is finite (Assumption 3.1. ii) let us assume that

P 1

H(0). This means that 0 c lev H. Since lev H is weakly closed (Proposition 5.2). to prove weak compactness

13 P

we only need to show that lev H is weak sequentially compact (Eberlein-Smulian B

Theorem) and i n Pn(a) this actually turns out to be the same as stmng 1 precompactness.

We first intend to show that lev H is locally weak sequentially compact at 0. To

P

see this, consider the continuous linear functional

where for a l l t, et = (1.1.

...,

1). Note that te, x> = llxll whenever x c dom H. Now le t

V: = lev H f l {x

1

(e, x > i 1)

P

This is a closed weak neighborhood of 0 relative to lev H. Pick any sequence

P

{x V ,v= 1,

...I

C V; we must exhibit a convergent subsequence characterized by (5.14). If some subsequence converges i n norm to 0, there is nothing

to prove, so we suppose that for some

>

0

u u

Therefore by a standard diagonal pmcedure we can extract a subsequence {x: u c N*) such that

Next we pmve that the set lev

P

is norm-bounded. Suppose to the contrary that there exists a sequence (xu E lev$(, u=l, ...) such that l i m llxYl =

+=.

^Define

m

yu: = xu / Ilx II u

assuming that llxu II

>

⁰ for a l l u. Since llyull = 1,

yu c

v

for a l l u

and, passing to a subsequence i f necessary, there exists y = l i m yUas follows

V*

from the weak sequential compactness of V. Since the yV converge weakly to y, i n particular we have

V V

1 = l i m llx II = lirn (e, y

>

= (e, y > = Ilyll.

lw- ^V*

Since lev H is convex our construction would imply that 0 # y t r c lev H (with

B

B

yt 1 0 for a l l t). Then

for a l l X

1

0, using the fact that 0 ^t lev H. The convexity of h now yields

ti

1-a' 1-a T-1 t-1 t - 1

B 1

l i m sup - h(--T (x0 + k a a yt.

*

^{2 Y ~ ) ) (5.22)}

T* 1-a 1-a

As T goes to ^+QD, aT goes t o 0 and

is well defined since llyll ( QD, with z # 0 since y # 0. Dividing both sides of (5.22) by X and letting X go to ^QD, we obtain

0 = l i m A-ID

1

l i m A-I (I-=)-' h((l-a)xo

+

Xaz. Xz).

A* A-

which w i t h formula (3.8) and condition (3.6). a consequence of Assumption 3.1, implies

contradicting (3.9). Hence y must be 0, and thus lev H is bounded.

P

To complete the pmof it suffices t o observe that local weak sequentially compactness and boundedness yield weak sequential compactness.

o

5.4 THEOREM. Existence. Suppose P is proper and satisfies the s t r i c t version of Grinold's qrowth condition (Assumptions 3.1

and

3.3). Then the essential objective function F o f P is weakly inf-compact, and hence there exists optimal solutions o f P.

Since F is proper, the inf-compactness implies the existence of optimal solutions. a

The preceeding theorem suggests that we could restrict ourselves t o trajectories that satisfy some uniform summability condition, but a t the outset we do not know the pertinent parameters and it would be inappropriate t o introduce them artificially. L e t us stress here the f a c t that these last results very much depend on having a

<

1, w i t h a = 1 we are i n another ballpark and the rules of the game are then quite different.

Im Dokument Existence Results and Finite Horizon Approximates for Infinite Horizon Optimization Problems (Seite 29-43)