Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
1901
I
snoitcnufitluM dna sdnargetnI
Stochastic Analysis, Approximation and Optimization
Proceedings of a Conference
held in Catania, ,ylatI June 7-16, 1983
Edited by .G Salinetti
IIIII
galreV-regnirpS
nilreB Heidelberg New York oykoT 1984
Editor
Gabriella Salinetti
Dipartimento di Statistica, Probabilit& e Statistiche Applicate
UniversitY. di Roma "La Sapienza"
00100 Roma - Italy
AMS Subject Classification (1980): 60-XX, 49-XX, 93 EXX, 52A 22, 28-XX
ISBN 3-540-13882-X Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN 0-387-13882-X Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging in Publication Data. Main entry under title: Multifunctions and
integrands. (Lecture notes in mathematics; )1901 Sponsored jointly the by Dipartimento di
statistica, probabilit& e statistiche applicate of the Universit~ Sapienza" "La di Roma andt he
Seminario matematico, Universit& di Catania. Bibliography: .p .1 Stochastic analysis-Congres-
ses. .2 Approximation theory-Congresses. 3. Mathematical optimization-Congresses. ,I Salinetti,
G. (Gabriella), 1946-. LI Universit& degli studi di "La Roma Sapienza": Dipartimento di statistica,
probabilit& e statistiche applicate. III. Universit& di Catania. Seminario matematico. .VI Series:
Lecture notes in mathematics (Springer-Verlag); .1901
QA3.L28 .on 1901 015 s 84-23565 [QA274.2] [519.2]
ISBN 0-387-13882-X ).S.U(
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PREFACE
This Dublication presents the Proceedings of the international conference
on "MULT!FUNCTIONS AND IN~RANDS: Stochastic Analysis, Approximation and Opti-
mization" held in Catania, Italy, in June 1983 under the scientific direction of
R.T. Rockafellar, .M Valadier and .G Salinet~i.
The purpose otfh e conference was to survey the current state of the
art, report on recent progress and delineate further directions of research with
special attention to applications.
Modern applications in statistics, probability, operation research, physics
and economics have focused attention on situations and mathematical models
where solutions aren ot necessarily unique, feasible sets depend - oftemne asu-
rably - on parameters, perturbations play a crucial role, functions have lost
their "traditional" smoothness up to the inevitable replacement otfh e notion of
continuity with the notion of semicontinuity.
This great variety of new situations and exigences has naturally led to
develop the theory of multifunctions and integrands as a new tool to deal with
them and the wealth of the results in the last decades rewards defJ_nitely this
approach. In this process that we feel at the height of its development the con-
ference hast rie@ to define where we stand and to indicate further directions of
research.
We did not try to organize the presentation of the papers
following the title of the conference: stochastic analysis, approximation and
optimization. Every paper is so rich in all these aspects that any classifica-
tionw ould have sacrificed part of its contents. So the papers follow
the alphabetical order of authors' names.The only exception is the paper by R.T.
Rockafellar and R.J-B. Wets Systems, '~/ariational an Introduction" Which opens
these Proceedings: ti seemed appropriate to begin with a comprehensive pre-
sentation containing motivations and fundaments otfh e theory as a guide to the
reader through the other papers. This introductory part corresponds to the tuto-
rial lectures that professors Rockafellar and Wets gave at the conference to
provide an introduction to the core of the main subjects and a guide towards the
most promising areas of research.
This conference was organized jointly by theD ipartimento di Stati-
stica, ?robabilit~ e Statistiche Applicate of the Universit~ "La Sapienza" di
IV
Roma and the Seminario Matematico of the Universit& di Catania with the finan-
cial support of C.N.R., E.N.I. and Universit& di Catania which contributed with
great interest. I must single out Giorgio Dall'Aglio (Universit~ di Roma), Car-
melo F~a (Universit~ di Catania) for their continuous assistance and encoura-
gement and Roger Wets for providing such elbaulavni expertise in the scientific
organization of the conference. Thanks also to the menders of the local organi-
zation committee, professors Santagati, Motta, Ricceri , Villani and to prof.
Chiarenza for so highly contributing to the preparation and the running of the
meeting.
Rcma, May 18th 1984
Gabriella Salinetti
CONTENTS
R.T. RCCFAFEIIAR- R. J-B. WETS, Variational Systems, an Introduction ....... I
X.ARKIN, V Extension of the Class of Markov Controls ......................... 55
.Z ARTSTE2N, Limit Laws for Multifunctions Applied to an Optimization
Problem ..................................................................... 66
H. ATIDUCH, Variational Properties of Epi-convergence. Applications to
Limit Analysis Problems in Mechanics and Duality ~heory ..................... 80
J.P. AUBIN, Slow and Heavy Viable Trajectories of Controlled Problems.
Smooth Viability Domains .................................................... 105
.C CASTAING, A New Class of Evolution Equations in a Hilbert Space .......... 117
1
A. CELLINA, A Fixed Point Tneorem for Subsets of L ......................... 129
N. CRESSIE, Modelling Sets .................................................. 138
E. DE GIORGI, On a Definition of F-convergence of Measures .................. 150
F. HIAI, Strong Laws of Large Numbers for Multivalued Random Variables 160 ......
T.G.KURTZ, Approaches to Weak Convergence .................................... 173
A. MARINO - M. DE GIOVANNI - M. TOSQUES, Critical Points and Evolution
Equa.t.i.o.n.s. ............................................................. 184
C. OLECH, Decomposability sa a Substitute for Convexity ....................... 193
J.E.SPINGARN, Multifunctions Associated with Parametrized Classes of
Constrained Optimizat ion Pr.o.b.l.e.m.s. .................................... 206
L. ~}{IBAULT, Continuity of Measurable Convex Multifunctions ................. 216
.M VALADIER, Some Bang-Bang Theore~ns ........................................ 225
LANOITAIRAV ,SMETSYS NA NOITCUDORTNI
R.T. Rockafellar m dna Roger J-B. steW ~
Mathematics EDAMEREC
University of Washington Universit~ Paris IX
Seattle, AW 98195 F-75775 Paris Cedex 61
1. SELPICNIRP LANOITAIRAV DNA .STNIARTSNOC
Fundamental in many applications of mathematics is the idea of modeling a situa-
tion yb first describing a set S of possible "states" that need to eb considered dna
then introducing additional criteria that single out from S some particular state
x . For example, S could represent all the configurations that might eb taken on yb
a certain physical system, dna x could eb na "equilibrium" state, perhaps expressing
a balance of forces or giving na extremal value to some energy function. Economic -om
dels often follow a similar pattern, except that instead of na energy function it yam
eb a cost or utility function, say, whose minimum or mumixam puts the spotlight on a
particular x in S . Such models too can concern an x which is na equilibrium re-
sulting from interactive maximization or minimization of various functions yb suoremun
individual agents.
Modern applications in statistics, engineering, and operations research have
especially focused attention no situations where a physical or economic system nac eb
affected or controlled by outside decisions, dna these decisions should eb taken in
the "best" possible manner. ehT notion of na optimization problem sah proved very use-
ful. In abstract terms, such a problem consists of a set S whose elements, called
the f~i61e solutions to the problem, represent the alternatives that are open to a
decision maker. Examples of S include the set of acceptable estimators for a statis-
tical parameter, the set of feasible designs in a structural engineering problem, the
possible control policies for na inventory process, dna os on. ehT aim is to minimize
over S a certain function f , the objective function. ehT elements x of S where
the minimum is attained are called the op~maZ soZu~o~ to the problem. Of course
minimization could eb replaced by maximization.
Supported in part by a grant of the National Science Foundation.
~ • nO leave from the University of Kentucky ; supported in part by a Fellowship of
the Centre National de la Recherche Scientifique.
In all such cases where an x is singled out from an underlying set S on the
basis of some kind of minimization or maximization, it is common to speak of x as
being characterized by a ZanoZt~oL~yov /y~LncZpZe. This terminology also carries over to
many situations where x does not necessarily give a true extremum but merely satis-
fies conditions that generalize, or form part of, various conditions known to be
associated with an extremum over S .
The question of variational principles and their role in science and technology
is closely connected, therefore, with understanding and characterizing extremals of a
function f over a set S . This in turn depends on the nature of f and S , and
here it is that a great amount of mathematical innovation has become necessary in re-
cent decades. The older view of variational principles was too limited. Traditional
methods are simply not adequate to treat the kind of functions f and sets S that
that nowadays are deemed important in such a context. eW speak here not just of spe-
cial techniques but of the entire outlook of classical analysis.
To begin with, some words about the sets S that may be encountered will make
this clearer. In this introduction, we shall be concerned in the main with situations
that can be described by a finite number of real variables, or in other words, which
display "finitely many degrees of freedom". Denoting the variables by I x ..... n x , ew
can identify the possible "states" which correspond to a situation at hand with ele-
ments (x I ..... Xn) = x of the space R n. Thus the state set S can be thought of
simply as a certain subset of R n. The exact definition of S in a particular case
depends of course on various circumstances, but it typically involves a number of
functional relationships among the variables I x .... n x . It yam also involve restric-
tions on the values that yam be taken on by these variables, In economic models, for
instance, it is common to have variables that are intrinsically nonnegative ; in
structural design problems, bounded variables are the rule.
A great many situations are covered by the following kind of description :
(1.1) S := set of all x = (x I ..... nX ) n c R such that
EX and fi(x) I ~0 for i = i ..... s,
x
=L 0 for i = s+l ..... m,
where X is some given subset of n R (usually rather simple in character, perhaps the
entire space n R ) and each fi is a real valued function on n R . The conditions
x EX , fi(x) ~0 or fi(x) = 0 are called consty~nCs on the state x . The inclu-
sion of the abstract condition xE X allows an open-ended flexibility in the des-
cription of the constraints.
What most distinguishes the applications for which classical analysis was deve-
loped from the modern ones, as far as sets S of type (I.I) are concerned, are the
inequality constraints, possibly very many of them, dna the frequent lack of
"smoothness" of the functions fi dna set X . In elementary models for physical
systems, it is frequently the case that S is completely characterized by several
equations involving the variables I x ..... n x :
(1.2) S = {x E X I fi(xl ..... x n) = 0 for i = I ..... m},
where X is na open set in n and R the functions fi are smooth, i.e. continuously
differentiable. Furthermore, the equations are independent in the sense that in a
neighborhood of any point of S they nac eb solved for emos m variables sa smooth
functions of the other m-n variables, although just which ones might depend on the
point in question. Then S is a "smooth" curve, surface, or hypersurface in n R of
dimension m-n , the kind of object which finds its abstraction in the important -am
thematical concept of a differentiable manifold. eW refer to such na S sa a smooth
manifold.
nehW inequality constraints are encountered in classical analysis, they are
usually of na elementary sort dna few in number. nA example of a set S that nac eb
described in terms of such constraints is a closed annulus : a region in 2 R lying
between two concentric circles and including the circles themselves. This corresponds
to two quadratic inequalities. Another example is a solid cube in 3 R or its bounda-
ry. Such a cube nac be determined by a system of six linear inequalities. Note that
nehw S is such a cube, its boundary is not a smooth manifold, but its structure is
simple enough not to pose hcum trouble. ehT open faces dna edges of S are smooth
manifolds that nac eb investigated individually. In general, one might say that the
kind of sets S seen in traditional applications are, if not smooth manifolds them-
selves at least the union of a modest number of smooth manifolds that are nicely
juxtaposed to each other dna easily listed in na explicit manner.
In contrast, many contemporary problems in economics, chemical equilibrium, phy-
sical variational principles, dna other areas, concern sets S of the form (i.i)
where the number of inequality constraints is in the hundreds or thousands, far lar-
ger than the number of variables i x , which nevertheless nac eb huge too. Then the
notions and technical tools appropriate for smooth manifolds on longer suffice. At
any given point x of S some of the inequality constraints can eb active (satis-
fied sa equations), while others nac eb inactive (satisfied with strict inequality).
Quite apart from the large numbers involved, there is usually no easy yaw to determine
which combinations of active dna inactive constraints actually do occur; cf. Figure I.
Furthermore, the consideration of such combinations does not necessarily lead to a
decomposition of S into smooth manifolds, not to speak of eno having a simple, di-
rect description. Even the equality constraints appearing in (I.i) nac cause diffi-
Figure 1. S the set of acceptable states.
culties by not being "independent" at critical points of S , dna the set × yam
have complicated boundaries that need to eb taken into account.
ehT study of evolutionary systems in the context of viability theory, cf.
Aubin, 1984, obliges su to confront, in a dynamical setting, all the questions that
were raised in connection with the mathematical structure of the set of acceptable
states sa defined by (i.I). ehT motivation comes from biological, ecological dna
macro-economics models that fit the following general evolutionary format : a closed
subset S of n R identifies the acceptable states of the system, the dynamics of the
system are described by the relations
(1.3) x(t) E r(t) dna X(to) = x ,O
where x denotes the derivatives of the state x with respect to t (the time
parameter), dna F(t) is the set of feasible dynamics at time t . In the study of
the evolution of the state x(t) sa a function of t , ew must make provisions for
the behaviour of the system at its frontier of viability, i.e. when x(t) belongs to
the boundary of S . Because precisely these critical periods are the ones of inte-
rest in the modeling process, ew cannot resort to the "smooth" case studied in clas-
sical dynamics, i.e. when the system is to evolve in na open domain or on a smooth
manifold with open boundaries.
Another difficulty is that the differentiability assumptions or differential
dependence of the solution no the parameters of the problem which seem (or at least
used to seem) os natural in classical physics lose their luster in other subjects.
Mathematical models derived from biology, economic theory or the theory of extremals
in statistics, for example, often have a convex set X dna inequalities involving
conve× functions fi ' These particular mathematical properties are of interest
because they have na axiomatic significance in economic models or extremal statistics
which smoothness properties od not. This turns out to eb on impass for analysis, if
certain generalizations of differential calculus are pursued.
ehT importance of being able to work with nonsmooth functions comes from more
reasons than just this. In emos way, inequality constraints in themselves force the
considerations of nonsmoothness. eW have already observed this in the example of a
solid cube in 3 R having a nonsmooth boundary. More generally, any constraint sys-
tem of the form gk(x) ~0 , for k = 1 ..... q , can be lumped together sa a single
inequality g(x) ~0 where
(1.4) g(x) : xam gk(x)
k=l, .... q
ehT price to eb paid, of course, is that g will not inherit the differentiability
properties of the functions gk ' see Figure 2.
gl
Figure 2, The max-function g.
Nevertheless the idea of lumping constraints together this yaw sah its value, dna ew
must be prepared to cope with it. For example any convex function g n : ~ R R can
eb represented by a limiting version of (1.4) in which infinitely many (linear) func-
tions are allowed.
ehT classical approach to a nonsmooth function g sa in (1.4) would eb to treat
it sa a piec~w~e smooth, or in other words to decompose the domain of g into fini-
tely many smooth manifolds relative to which g is continuously differentiable. But
this yam eb impossible without imposing painful dna practically unverifiable conditions
on the functions gk dna woh they interact with each other.
ssenhtoomsnoN enters the study of variational problems through the analysis of
