Stochastic Lipschitz functions - Springer Link

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M. E. Primak, "On convergence of the cutting plane method with cleaning at ... A single-valued map g:O-+R ~ is called a section of the multivalued map O:~--~ n,.
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9. 10. 11.

G. Dantzig, Linear Programming, Generalizations and Applications [Russian translation], Progress, Moscow (1966). P. Huard, "Resolution of methematical programming problems with nonlinear constraints by the method of centers," in: Nonlinear Programming, North-Holland, Amsterdam (1967), pp. 206-219. J. Kelley, "The cutting plane method for solving convex programs," SlAM J., 8, No. 4, 703-712 (1960). M. E. Primak, "On convergence of the cutting plane method with cleaning at each step," Kibernetika, No. I, 119-121 (1980). E. S. Levitin and B. T. Polyak, "Methods of constrained minimization," Zh. Vychisl. Mat. Mat. Fiz., 6, No. 5, 787-823 (1966).

STOCHASTIC LIPSCHITZ FUNCTIONS V. I. Norkin

UDC 517.987.1

A successful generalization of the concept of the gradient was found for Lipschitz functions by Clark [I]. Gupal [2] developed finite-difference methods for the minimization of locally Lipschitz functions, not using Clark's generalization of gradients, but converging to stationary points of these functions in the sense of Clark. In the present paper we study the Clark subdifferential of random Lipschitz functions. We show that it is a multivalued mapping which is measurable in the collection of determinate and stochastic variables. (The information needed from the theory of measurable multivalued mappings is given in the paper.) The measurability of the Clark subdifferential with respect to a random variable, which is necessary for its integration, is proved. A formula concerning the subdifferential calculus is proved in general. It is shown that the subdifferential expectation of a stochastic Lipschitz function is included in the expectation of the subdifferential of this function (as the integral of a multivalued map). Under assumptions similar to convexity, such an inclusion is established in [3]. We note that for convex functions the Clark subdifferential coincides with the ordinary subdifferential of a convex function (the set of subgradients). Thus, we have an inclusion of the subdifferential of the expectation of a stochastic convex function in the expectation of its subdifferential. The opposite inclusion follows trivially (under the measurability of the subdifferential which has been proved already) from the subgradient inequality for convex functions. Thus, in the convex case we have a familiar result: the equality of the subdifferential of the expectation and the expectation of the subdifferential. In [4] one can find a proof of this fact for functions on abstract spaces as well as information of historical and bibliographical character. Measurable Multivalued Mappings In what follows we shall need some information from the theory of measurable multivalued mappings [4-I J]. Let O be a set, Z be a o-algebra of measurable subsets of O, Rn be n-dimensional Euclidean space, 2Rn be the set of subsets of Rn, ~ (or ~Rn) be the o-algebra of Borel subsets of Rn, ~ X ~ be the o-algebra in Rn • O, generated by all subsets of the form B • T, where B E ~ , TEE. 2.

By multivalued map from O

to Rn is meant any map 0:O--~2 ~, doing----{0EOIO(0)~=~}.

3. A single-valued map g : O - + R ~ if g(0)EO(0) for all 0CdomO. 9

is called a section of the multivalued map O : ~ - - ~ n,

Rn

4. A multivalued map G : O ~ + 2 is called measurable, if domG is measurable and there exists a countable family of measurable sections, gl:domG § R n, I = I, 2,..., of the map G, such that the values {gz(0)}z~=l are dense in G(0) for all 0EdomO. 5. A multivalued map 0:0.-+9 Rn is called normal, if it is measurable and G(0) is closed for all 0EdomO. Translated from Kibernetika, No. 2, pp. 66-71, 76, March-April, 1986. submitted November 11, 1983.

226

0011-4235/86/2202-0226512.50

Original article

O 1986 Plenum Publishing Corporation

6.

THEOREM ] (Castaing [5], Rockafellar [7]).

The following assertions about the

multivalued map O:O.->-9.a" are equivalent: I) G is normal; 2) G(8) is closed for all 0 6 d o m B~ Rn

O

and for any closed set (or for any compact set)

O-~ (B) = {06010(0) N B=/= J~}6 ~,. n • O--~ RiU {q- oo} will be called an integrant.

7.

Any function r

8.

We denote by ~00(x) the function

epi r

=

x-+~(x, 0);

{(x, =) 6 R ~+1 [ = ~> ~0 (x)}

is its supergraph. 9. The integrant is normal.

~'Rn • O'--~-Ri O {q- oo} is called normal if the multivalued map 0.-~epi~0

10. The integrant ~ : R n • vex for all e.

{q-o o} is called convex if the functions ~e{x) are con-

11. LEMMA I (Rockefeller [7]). then the multivalued map

0-~

If ~p:Rn X O - + R t U { q -oo}

is a normal convex integrant,

(0)= {xER"l~(x,0)~ ~ (g, x, 0)} is ~ X ~

-measurable and the sets

epi~,0 are not empty and are closed for all (x, 0).

18. Suppose there is defined on the space (O, Y.) a o-finite positive complete measure P. For this case, in all the definitions given above, and also in Theorem I and Lemma I, the condition "for all 0" can be replaced by the condition "for almost all 0." In the case considered, for a closed-valued map G : O - ~ 2 one can g~ve many other equivalent definitions of measurability (cf. [4-6, 8, 10]). Lemma 2, Proposition ~, and Corollary I are somewhat changed. 19.

We shall call a map g:Rn• m ~XY-almost measurable, if there exists a set such that g:R n XO'__,.R 'n is ~ X F . . ' -measurable, where the o-algebra E' =

O'6Y., P ( O ~ O ' ) = O

{T fl O'ITE~}c ~. 20. If g : R ~ X O--~-R ~ is ~)~n X ~' -almost m e a s u r a b l e , and h : R l • O - ~ R ~ is ~Rz X ~ almost measurable, then the composite map g (h (x, O),8) "R t X O - ~ R ~ is ~R~ x y, -almost measurable. 21. We shall call the multivalued map G : R n x O - ~ p.R~ ~ X E -almost measurable if for some 0 ' 6 E , p ( O ~ O ' ) = O , the map G:RnX6), _~pflm is ~ X P.' -measurable, where the o-algebra

Z' = { r N O'lVE~IcZ22. The map g : R n x O - ~ R m is said to be composition measurable [10] if for any measurable map x : D ~ R n , the composite map g(x(O),8):O--~R ~ is measurable. 23.

If g : R n •

is ~ X ~

-almost measurable,

then g is composition measurable.

24. LEMMA 2'. If ~ : R ~ X O - + R i U { + o o } is a normal integrant, then the function ~ , ~ is ~ X ~ -almost measurable and, consequently, composition measurable [10]. 25. Proposition I' A Caratheodory function f:R" X O - + R I is a normal consequently ~ X E -almost measurable and composition measurable [10]. 26. COROLLARY |' A Caratheodory map composition measurable [10].

integrant, and

g: R n X e~-~R m is ~ X ~. -almost measurable and

27. In [10], Lemma I, the composition measurability of normal integrands and Caratheodory maps is also proved without the assumption of completeness of the measure P. 28. A multivalued map G:Rn § R m is said to be upper semicontinuous and for any x 6 R n one can find a ~ > 0 such that for all g6G(y),

if for any g > 0 one has

Iiy--x[] 0 and

IIg~(o)--g~ll+8}.

inf

gxEO(x,O)

as follows:

+ g'~ (0), 0 ~ 0~, I.g~(o), oe,:,+, . ,-~2\ OI,, tn~I

{ C (o), oc 0 m+ \ U

e~' (o) =

0 +, ~

i=l rn

o c o \ U o;. Obviously,

the functions

gm are measurable

sections of G(x,

9), while on the set

6 0 i8 I=I

IJgy (0) -- g~ (0) [I~< inf IIgv (0) -- gx II + s. gx66(x,O)

We show that the measure /~e = P

O~

O

-+0, m -+oo.

In fact,

O

is a monotone

increasing

i=l

9

oo

m e a s u r e Pe ~ 1.

If

:/

I, so the limit set U o; is-measurable and its kill / oo p o o > o , for all m one has Pe < 1, then on the set o ~ U o ~ ,

sequence of measurable

sets, while

P

O

IIg, -- gT~ ~')) II>

inf

gxEGIx.Ot

IIg, (0) -- gx II +~oo

This means that the family

{gxm(0)}~ffil on the set o ~ g o t e

is not dense in G(x, 0).

tradiction found proves that the measures /9~ = p ( o ~ U o

! Ilgy (0)-- g~, (0) liP(dO) ~
[f (x + y + Xd, 0) - - f (x + y, 0)]/L. !

The functions f~(x, e; d) and f'(x, O; d) are measurable with respect to e (this is proved in exactly the same way as their ~ X Y. -measurability in Lemma 4) and since f~(x,0;~