Computing Methods in Optimization Problems. Proceedings of a Conference Held at University of California, Los Angeles January 30–31, 1964

322 Pages·1964·11.597 MB·English

COMPUTING METHODS
IN OPTIMIZATION
PROBLEMS
EDITED BY
Α. V. Balakrishnan
UNIVERSITY OF CALIFORNIA
LOS ANGELES
Lucien W. Neustadt
AEROSPACE CORPORATION AND
UNIVERSITY OF MICHIGAN
Proceedings of a conference held at
University of California, Los Angeles
January 30-31, 1964
1964
ACADEMIC PRESS New York and London
COPYRIGHT © 1964, BY ACADEMIC PRESS INC.
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NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
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PRINTED IN THE UNITED STATES OF AMERICA.
PREFACE
The impact of high-speed computers on optimization problems
has been well recognized and indeed the phrase 'computer revolution'
is now a cliche. That the need is not so much for existence theory as
for efficient computing algorithms is no longer disputed. Nevertheless
our knowledge of specific computing methods for such problems is still
sparse. Much extant theory on direct methods for instance is still
based on experience with hand-computers. The conference on Comput­
ing Methods on Optimization Problems was organized to provide a fo­
rum for the dissemination of recent research progress in the area and
in particular to share computing experience gained in specific large-
scale problems.
This book is based on the papers presented at the Conference.
Owing to the press of time in trying to meet an early publication date it
has not been possible to include the Discussion contributions. It was
also found that it would be inefficient to rearrange the papers further
by subtopics. The lone exception is the group of the final three papers
on 'hybrid computing methods' which is separable as a unit under this
heading. We have therefore merely followed the conference sequence.
It is our pleasant duty to acknowledge the active cooperation and
help of the UCLA Department of Engineering and Chairman Duke in the
preparation of the book. We would also like to thank Academic Press
for its invaluable assistance throughout.
Α. V. Balakrishnan and
L. W. Neustadt
ν
LIST OF CONTRIBUTORS
Page
Bekey, George Α., Electrical Engineering Department,
University of Southern California 305
Bellman, R., The RAND Corporation 135
Brunner, Walter, Electronic Associates, Inc . 285
Fadden, Edward J., Instrumentation Engineering Program,
The University of Michigan 167
, ,
Faulkner Frank D. Department of Mathematics and
Mechanics, U. S. Naval Postgraduate School 147
Gilbert, Elmer G., Instrumentation Engineering Program,
The University of Michigan 167, 261
Goldstein, Α. Α., The University of Texas 159
Halbert, Peter, Electronic Associates, Inc 285
Halkin, Hubert, Bell Telephone Laboratories 211
Hestenes, Magnus R., Department of Mathematics,
University of California Los Angeles 1
,
Hillsley, R. H., International Business Machines
Corporation, Federal Systems Division,
Space Guidance Center 107
Hsieh, H. C, Department of Electrical Engineering,
Northwestern University 193
Kalaba R. The RAND Corporation 135
, ,
Kopp, R. E., Research Department, Grumman Aircraft
Engineering Corporation 65
McGhee, Robert B., Electrical Engineering Department,
University of Southern California 305
McGill, R., Research Department, Grumman Aircraft
Engineering Corporation 65
vii
Page
Mover, H. Gardner, Research Department, Grumman
Aircraft Engineering Corporation 91
Paiewonsky, Bernard, Aeronautical Research
Associates of Princeton, Inc 285
Perret, R., Faculté des Sciences, Grenoble, France 241
Pinkham, Gordon, Research Department, Grumman
Aircraft Engineering Corporation 91
Robbins H M. International Business Machines
Corp,orat.ion, ,Federal Systems Division,
Space Guidance Center 107
Rosenbrock, H. H., Cambridge University* 23
Rouxel, R., Batelle Memorial Institute 241
Storey, C, I.C.I. Central Instrument Laboratory 23
Woodrow, Peter, Aeronautical Research
Associates of Princeton, Inc 285
* Present address - Massachusetts Institute of Technology
viii
VARIATIONAL THEORY AND OPTIMAL CONTROL THEORY*
by Magnus R. Hestenes
University of California
Department of Mathematics
Los Angeles, California
1. Introduction. The problem to be considered is that
of minimizing a function of the form
- ,1
l = g^w; + / i^,x^;,u^;,w; αχ
in a class of functions and parameters
xX(t), uk(t), wa
(t°^t^ t1; i=l, . n; k=l, ..., q; σ=1, r)
satisfying the differential equations and auxiliary conditions
x1 = tx(t,x,u,v) (1:1a)
tS = TS(w), xi(t ) =XiS(w) (s=0,l). (1:1b)
s
q>(t,x,u,v) S 0 (lêaSm'), 9(t,x,u,w) = 0 (m'<aSm) (l:lc)
a a
y 0 (lS7Sp«), I = 0 (ρ·<7«ρ)
7
where
t1
(w) + f f (t,x(t),u(t),w) dt (1:2)
τ7 = g7 J 0 7
In addition one may have constraints of the form
•(t,x,v) * 0 ( Ι ί β ^) (1:3)
p
The preparation of this paper was sponsored by the Office
of Naval Research and the U. S. Amy Research Office (Durham).
Reproduction in whole or in part is permitted for any purpose
of the United States Government.
1
COMPUTING METHODS IN OPTIMIZATION PROBLEMS
We have omitted conditions of the foim
*p(t,x,v) = 0
in (1:3) since these can be replaced by the conditions
Vp(t'x'u'w> - v + * ifi = 0
px
= ψβ tT°( vh χ10ΜΛ -0
which are of the form (l:lc) and (l:ld). It should be observed
that there is no generality of including w in f, f , φ ,
y a.
σ
since the parameters w in these functions may be replaced by
new variables χη+σ subject to the conditions
.η+σ η+σ,.Ον η+σ,.ΐν σ
χ = 0 , χ (t ) = χ (t ) = w .
We have included wa in these functions in order to indicate
how they enter into the multiplier rule.
The inequality constraints
9(t,x,u,w) £ 0 1-°
a 7
can be replaced by the constraints
Φ (u**)2 =0 I {yl^f - 0
α + y +
Q +Q! r+*y
in which u are additional functions and w are addi­
tional variables. This method is due to Valentine (l). We
shall not make this replacement, since it is unnecessary to do
so in deriving first-order necessary conditions. It is, how­
ever, convenient to use this device in the derivation of
second-order conditions.
We shall not discuss the case in which constraints of the
form (1:3) are present. Problems of this type have been dis­
cussed by Berkowitz (2), Gamkrelidze (3), and more recently by
Guinn (k). In the development of this theory one uses uni­
lateral variations. Elementary cases have been treated by
Bliss and Underhill (jj) and others.
The problem formulated above is a very general problem
in the calculus of variations and is equivalent to the problem
of Bolza, as we shall see presently. It contains as special
cases a large class of optimal control problems arising in
2
COMPUTING METHODS IN OPTIMIZATION PROBLEMS
applications. We shall accordingly refer to this problem as
the optimal control problem in variational theory. The vari-
k σ
ables u will be called control variables, the parameters w
will be called control parameters, and the variables x1 will
be called state variables. The first general fonnulation of
problems of this type known to the author was that given by the
author (6) in a Rand report concerned with paths of least time.
This report was not published in a journal. In this report
constraints of the foira. (l:ld) and (1:3) were not considered.
The technique employed would admit constraints of the form
(l:ld) but would have to be modified to include those of the
form (1:3).
The general problem of Bolza referred to above is that
of minimizing a function of the form
t1
= g(w) + f f(t,x(t),x(t),v) dt
Ao
in a class of arcs and parameters
xi(t), w° (t° £ t £ t1; i=l, . . . η; σ=1, ...y r)
y y
satisfying the relations
φ (t,x,x,w) ^ 0 (l^a^m1), q>(t,x,i,w) = 0 (m^a^m) (l:ta)
a
tS = TS(w), x^t8) = Xis(w) (s = 0,1). (l:*rt>)
These arcs may be subject to constraints
I * 0 (ΐ^Ρ'), I = 0 (pf<7^p) , (like)
7 7
where
I = gjv) + / f (t,x(t),i(t),v) dt.
77 J 0 7
In addition one may have constraints of the form (1:3)· This
problem with inequality constraints excluded is equivalent to
one studied by Bliss (7) in his book on the calculus of vari­
ations. The case in which inequality constraints occur in
(l:l) can be reduced to the case in which no inequality con­
straints occur. This can be done by the method of Valentine
described above. To account for conditions of the form (1:3)
one can use the theory of unilateral variations.
The problem of Bolza as formulated above reduces to an
optimal control problem by considering
3
COMPUTING METHODS IN OPTIMIZATION PROBLEMS
• I i
χ = u
to "be the control variables. Conversely, the optimal control
problem can be reduced to one of Bolza type by introducing new
n4"k r-i-k
variables χ , w connected by the relations
**() /•* k() , n +V) =w^k
x t = u B d s X
and eliminating u by the substitution
k .n+k
u = χ
It follows that the two problems are equivalent and that the
theory for one can be deduced from the theory for the other.
The author prefers the optimal control formulation to that
given by Bliss because it appears to be more convenient for
applications and because it leads to an approach to variation­
al theory that is both novel and instructive. The present pa­
per is concerned with this approach.
In 1936 Bliss (8) wrote a paper entitled "The Develop­
ment of Problems in the Calculus of Variations." In this pa­
per he traced the development of the Problem of Bolza from the
time of Euler and Lagrange. He pointed out that both Euler
and Lagrange studied variational problems of generality com­
parable to the one here considered. He made a strong case for
the study of the problem of Bolza as a standard problem in
variational theory. However, in his evaluation he did not
consider the optimal control problem here formulated.
Professor Bliss was the leader of the Chicago School in
Variational Theory. L. M. Graves, W. T. Reid, E- J. McShane,
and the author, together with their students, were active mem­
bers of this school. This group together with Marston Morse
developed the theory for the Problem of Bolza to a state of
completion comparable with those of simple problems. Most of
the known results for the optimal control problem are readily
deducible from known results for the Problem of Bolza.
At the time that I formulated the general control prob­
lem here considered and obtained the corresponding multiplier
rule there was very little interest in the problem. Several
years later it became evident that this problem was of great
importance in control theory. Consequently it has been stud­
ied intensely both here and abroad. Two principles have
evolved that have become popular, namely the maximum principle
of Pontryagin (9) and the principle of optimality of Bellman (10).
4
COMPUTING METHODS IN OPTIMIZATION PROBLEMS
In each case they represent the first-order necessary condi­
tions for an optimal solution. By first-order conditions are
meant those derivable "by the use of first derivatives, namely
the Euler-Lagrange equations, the transversality condition and
the Weierstrass condition or their equivalents. The maximum
principle of Pontryagin can be derived under weaker hypotheses
than have been used heretofore. The optimality principle of
Bellman can be looked upon as an extension of Hamilton Jacobi
theory, and reduces to this theory when applied to classical
variational problems.
The technique used by Pontryagin is a modification of
one used by McShane (11)· They obtained the first-order neces­
sary conditions by finding a hyperplane of support to a suit­
ably chosen convex cone. McShane used both strong and weak
variations to construct the cone. Pontryagin used only strong
variations. The modification of Pontryagin is significant be­
cause it enables one to weaken the hypotheses made. Moreover,
inequality constraints can be treated more readily. There is
a further modification of these methods which will simplify
the arguments, particularly when inequality constraints are in­
volved. It is our purpose to give an outline of this modified
approach to the establishment of the first-order conditions.
The details of the program describedbelow were given in a semi­
nar on variational theory at the University of California, Los
Angeles.
2. First-order necessary conditions for a local minimum.
Let
x: x^Ct), uk(t), wa ( t ^ t ^ t1)
0 0
be a solution to the optimal control problem in which the con­
straints (1:1) hold. As remarked above, we shall omit the dis­
cussion of the case when constraints of the forai (1:3) are
present. We shall assume that the control functions
UQ (t) are piecewise continuous for the purposes of our dis­
cussion. In addition we shall assume that the mx (q+m) di­
mensional matrix
(β not summed)
has rank m . Here & , = 1 , δ =0 (α φ β) . Under these
aQ
assumptions, together with the usual continuity and differenti­
ability assumptions, one can prove the following theorem. In