Table Of ContentLectu re Notes in
Operations Research and
Mathematical Systems
Economics, Computer Science, Information and Control
Edited by M. Beckmann, Providence and H. P. Kunzi, Zurich
56
Mohamed A. EI-Hodiri
University of Kansas, Department of Economics,
Lawrence, Kansas/USA
Constrained Extrema
Introduction to the
Differentiable Case with
Economic Applications
Spri nger-Verlag
Berlin· Heidelberg· New York 1971
Advisory Board
H. Albach· A. V. Balakrishnan· F. Ferschl . R. E. Kalman· W. Krelle . N. Wirth
AMS Subject Classifications (1970): 34H OS, 49Bxx, 90A 99
ISBN -13:978-3-540-05637-9 e-ISBN -13:978-3-642-80657-5
DO I: 10.1007/978-3-642-80657-5
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© by Springer-Verlag Berlin· Heidelberg 1971. Library of Congress Catalog Card Number 72-177567.
Softcover reprint of the hardcover 1st edition 1971
TABLE OF CONTENTS
Preface------------------------------------------------------------------------- 1
I. Introductory Survey
Chapter 1: A Survey of Derivative Characterization 0f Constrained Extrema-- 5
II. Finite Dimensional Problems
Chapter 2: Equality Constraints--------------------------------------------25
Chapter 3: Inequalities As Added Constraints-------------------------------34
Chapter 4: Extensions, and Applications-------------------------------------39
III. Variational Problems
Chapter 5: The Problem of Bolza with Equality Constraints------------------61
Chapter 6: The Problem of Bolza with Equality-Inequality Constraints-------78
Chapter 7: Extensions and Applications-------------------------------------84
PREFACE
These notes are the result of an interrupted sequence of seminars on optimiza
tion theory with economic applications starting in 1964-1965. This is mentioned by
way of explaining the uneven style that pervades them. Lately I have been using the
notes for a two semester course on the subject for graduate students in economics.
Except for the introductory survey, the notes are intended to provide an appetizer
to more sophisticated aspects of optimization theory and economic theory.
The notes are divided into three parts. Part I collects most of the results
on constrained extremf! of differentiable functionals on finite and not so finite
dimensional spaces. It is to be used as a reference and as a place to find credits
to various authors whose ideas we report.
Part II is concerned with finite dimensional problems and is written in detail.
Needless to say, my contributions are marginal. The economic examples are well
known and are presented by way of illustrating the theory.
Part III is devoted to variational problems leading to a discussion of some
optimal control problems. There is a vast amount of literature on these problems
and I tried to limit my intrusions to explaining some of the obvious steps that are
usually left out. I have borrowed heavily from Akhiezer [ 1], Berkovitz [ 7],
Bliss [lOJ and Pars [40J. The economic applications represent some of my work and
are presented in the spirit of illustration.
During the process of preparing for these notes, starting in Minnesota and end
ing in Kansas via Purdue, Missouri and Cairo, I have benefited a great deal from
my teachers and colleagues. I am most indebted to Professors: M. Beckmann, W. Brock,
A. Camacho, J. Chipman, M. El-Imam, L.S. Fan, C. Hildreth, L. Hurwicz, H. Kanemitsu,
D. Katzner, M. Khairat, W.S. Loud, J. Moore, J. Quirk, S. Reiter, R. Saposnik,
A. Takayama and R. Thompson. They must receive the wages of their sins of giving,
wittingly or unwittingly, implicitly or eXplicitly, aid and comfort to the author.
None of them, however, have seen the final draft of these notes. This, we hope, is
grounds for clemency. I alone am responsible for all errors.
2
The preparation of the final draft was done at the University of Kansas, and
I gratefully acknowledge the help and support that I received. Finally, it is a
pleasure to express my thanks to Mrs. Alice Milberger for the magical job of trans-
forming various bits and pieces of handwriting to a neat typescript.
Mohamed EI-Hodiri
Lawrence, Kansas
July 1971
"Now you shall hear the story of Sordello."
Browning: Sordello
I
INTRODUCTORY SURVEY
CHAPTER 1
A SURVEY OF DERIVATIVE CHARACTERIZATION
OF CONSTRAINED EXTREMA
1. Introduction
In this chapter we review the theorems that characterize optimality by way of
derivatives. First we formulate a very general optimization problem. Then we present
characterization theorems for three types of problems: Finite dimensional, varia-
tional and problems in linear topological spaces. In each case we present theorems
for equality - inequality constraints. The theorems in each case are: first order
necessary conditions, first order sufficient conditions, second order necessary con-
ditions and second order sufficient conditions. The scheme of representation is as
follows: Statements of theorems are followed by remarks referring the reader to the
earliest, known to us, proofs of the theorems. In some instances, slight generali-
zations of some theorems appear here for the first time, an indication of necessary
modifications to existing proofs are presented. A case is "solved" if proofs for all
the four types of characterization theorems exist. The only "unsolved" case is that
of problems in linear topological spaces with inequality and with equality - inequality
constraints. For this case, we present two conjectures about second order conditions
that are analogous to the equality constraint case.
We now state the "general" optimization problem:
Let A,B,C,D be ,real linear topological spaces. Consider the functions:
f: A + B
g: A + C
h: A + D
Let P be a partial ordering of B, let> be defined on C - as usual - by way of a con-
vex cone and let 81, 82, 83 and 84 be the neutral element of addition - zero elements -
1
of A,B,C and D respectively. The problem may be stated as follows :
lThis general statement of the problem is due to Hurwicz [27].
6
Find x e: A such that f(;{) is P-maxima12 subject to: g(x) = 93 and hex) 2:.. 94
In case B is the real line and P is the relation ">" defined for real numbers. we
have a problem of scalar optimization. In case B is finite dimensional we have a
finite-vector maximization problem. In general. characterizations of solutions to
finite-vector maximization problems may be derived from characterizations of solutions
of scalar maximization problems. We shall restrict our presentation to scalar maxi-
mization problems. As an application of scalar maximization theorems we may "solve"
a particular finite-vector maximization problem. namely for the case where P is taken
to be a Pareto3 ordering of B. For infinite-vector maximization problems. a scalari-
zation theorem appears to be the most appropriate intermediate step to derive charac-
terization theorems. Such a theorem was proved by Hurwicz [27J. The method. for
finite-vector maximization. consists4 of simply observing that the problem is equiv-
alent to a finite number of scalar maximization problems. However. we shall only be
concerned with scalar maximization problems in this chapter.
Finally. at the risk of blabbering the obvious. these things should be pointed
out. First. the choice of differentiable functions as the class to which the results
apply does not mean that we think them to be the most general class functions nor
does it mean that the proofs of the theorems are the most elegant. In fact. such
considerations are traded for unity and simplicity of the representation. Second.
the division of problems to finite dimensional. variational. and problems in linear
topological spaces. Had it been true that the problem of part 3 of this chapter is
"solved". there might be a case for disregarding sections 1 and 2. for then they
would be of only historical interest. But even then. one would be faced with the
problem of assuming too much in order to get the more general results and obtain others
as special cases. An illustration of the point is the relation between optimal con-
2We shall restrict our attention to maximization. Characterization of solutions to
minimization problems follow trivially from maximization theorems.
3minimizatio . ' .
=
In that case. x is said to be Pareto superior to y if flex) > fl(y) for i 1 •...• r
(r is the finite dimens ion of B). and ~ i_s Pareto_optimal (P:-maximal) if there does
not exist a point y e: A satisfying g(y) 2:.. 82, hey) = 93 which is Pareto superior to x.
4
Suggested to the author by Hurwicz, see chapter 4 of these notes.
7
trol theory, Part 2, and Part 3, functional analysis. There exist excellent works
relating the problem of maximizing a functional subject to functional constraints to
optimal control problems, see e.g. [llJ and [39J. We decided not to include these
papers in our survey because the results, on functional analysis, were rather special-
ized, i.e. they apply to functionals in general but they are more suited to variation-
al problems. One other reason for not dealing with variational problems as problems
in Linear Topological Spaces is that the theory, in Part 3, is incomplete, while vari-
ational theory is.
Third, since we are only concerned with characterization, computational results
and existence theory are not reported on.
2. Finite Dimensional Problems
Euclidian spaces of dimensions n, one, m and ~ respectively. For this part we reform-
ulate the scalar maximization problem as follows:
Problem 1. Find x £ Ern such that f(~) > f(x) for all x satisfying gU(x) = 0,
U = 1, ... , m, hS(x) ~ 0, S = 1, ... , ~
We further state some properties of the constraints that will be used in dis-
cussing the theorems in this section. Some of the nomenclature, designated by " ",
is Karush's [31J.
D.l) Definition. Effective constraints. Let ~ be a point that satisfies hS(~) ~ O.
S = 1, ..• , m. Let r(~) b.e the set of indices S such that hS(~) = o. The con-
straints with indices S £ r(~) will be referred to as the effective constraints
o
at x.
D.2) Definition. "Admissible Direction". Let h be differentiable. We say that A
is an admissible direction if A is a non-trivial solution to the inequalities:
n 013
i~lhi Ai ~ 0, 13 £ r(~)
where h~ xo .
l
D.3) Definition. "A curve issuing from ~ in the direction A." By that we mean, an
n-vector valued function, ~(t), of a real variable t such that ~(O) = ~ and
