Frontiers of Qumtitatiue Economics. Volume IT, ed. M. D. Infriligaforand D.A. Kerldrlck.
0 1974 North-Holland Publishing Company.
Applications of duality theory
en coordonnker ponctuelles, cer nouvelles relations pr6sentant l'avantage de fournir
explicitement h quantitks en fonction du revcnu et des prix.
Ainsi parait s'ouvrir un champ fkcond d'investigations pour les Cconombtres.
(Roy 1947, p. 225).
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CHAPTER 3
APPLICATIONS OF DUALITY THEORY *
W.E. DIEWERT
Department of Manpower and Immigration, Ottawa
Department of Economics, University of British Columbia
0. Introduction
There appear to be two principal practical applications of duality theory in
economics.
The first principal application of duality theory is that it enables us to
derive systems of demand equations which are consistent with maximizing or
minimizing behavior on the part of an economic agent, consumer or producer, simply by differentiating, a function, as opposed to solving explicitly
a constrained maximization or minimization problem. Perhaps the first
person to appreciate the econometric implications of the above statement in
the context of consumer demand theory was Rent Roy (1942):
Cette conception conduit k I'emploi de coordonndes tangentielles pour la dkfinition des
surfaces d'indifference et elle offre ainsi la possibilitk d'obtenir pour I'optimum du
consommateur, de nouvelles relations d'kquilibre, homologues des relations obtenues
* This research was partially supported by a Canada Council grant. In the five years
that 1 have been studying duality theory, many people have assisted me: S.N. Afriat,
E.R. Berndt, L.R. Christensen, M. Denny, L. Epstein, M. Fuss, Z. Griliches, R.E. Hall,
G. Hanoch, R. Harrls, D.W. Jorgenson, L.J. Lau, M. Nerlove, D. McFadden, P.A.
Samuelson, R. W. Shephard, R. M. van Slyke, E. Wiens and A. D. Woodland. The editorial assistance of M. lntriligator is also greatly appreciated. My thanks also to Lise Bldoo
for typing a difficult manuscript.Jhe views expressed are solely those of the author and
do not necessarily correspond to any policy or position of the Department of Manpower
a r ~ dImmigration.
One of the first authors to appreciate the usefulness of duality theory in
deriving systems of factor demand functions in the context of producer
theory was McFadden (1966, p. 13):
The introduction of cost, revenue and profit functions into production theory yields a
theoretical return and several practical advantages. Most of the qualitative results of
production theory follow from properties of these functions, without restrictive
assumptions on the divisibility of commodities and convexity and smoothness of
production possibilities.
The principal practical advantage lies in the simple relation between these functions and the corresponding demand and supply functions. For example, differentiation of the profit function yields net output supply functions, and summation.,of
prices times net output supply functions yields the profit function.
The second principal advantage of duality theory is that it enables us to
derive in an effortless way the "comparative statics" theorems originally
deduced from maximizing behavior.' This second principal advantage has
its origins in the work of Hotelling (1932, pp. 594, 597). McFadden (1966,
1970) has extensively pursued this second principal advantage.
In sections 1,2 and 3 of the present paper, we provide some examples of
the first principal advantage of duality theory, while sections 4, 5 and 6
provide some examples of the second principal advantage of duality theory.
In section 1, we study the duality between unit cost functions and constant
returns to scale production functions. The duality which is developed in
section 1 was known to SamuelsonZ(1953) and can be obtained by specializing Shephard's (1970) more general results. However, our present method
of developing the duality theorem is somewhat more direct than Shephard's
method, since we proceed directly from the cost function to the production
function via definition (8). In the appendix to this paper (section lo), we
outline a proof of a theorein which appears in the main text of the paper, ft
(i) the theorem contains a "new" result3 or (ii) the proof is new (and hope.
That the assumption of maximizing behavior implies certain restrictions on consumer demand functions was known to Antonelli (1968, pp. 33-39) (originally published
1886) and Pareto (1971, pp. 417-426) (originally published 1906), but the first rigorous
comparative statics result was obtained by Slutsky (1915). Hicks (1946) and Samuelson
(1947) systematically used the assumption of maximizing behavior in order to obtain
comparative statics theorems.
Samuelson does not provide a proof of the theorem.
Of course, many "new" results are straightfonvard modifications of known results.
.*
W.E. Diewert
f d y simpler than existing proofs). In any case, the reader who is primarily
interested in applying the existing body of duality theory can read the paper
independently of proofs, while the reader who is interested in modifying the
existing theory in order to suit his particular needs will probably find it
useful to peruse the proofs.
In section 7, we briefly survey the nonparametric methods of obtaining
systems of derived demand equations which are consistent with maximizing
behavior.
In section 8, we note how duality theory can make the variable coefficients
simplex algorithm as simple to use as the ordinary simplex algorithm for
linear programming.
From a mathematical point of view, duality theory rests on a theorem due
,
to Minkowski (191 1): every closed4 convex5 set in R~ can be characterized
as the intersection of its supporting half space^.^ We shall see how this theorem from mathematics creeps into economics in the following two sections.
Notation: x 2 0, means each component of the vector x is nonnegative;
ON is an N-dimensional vector of zeroes; x > ON means x 2 ON, but x # ON;
x D ON means each component of x is positive; E means "belongs to";
xTy is Zxiyi, the inner product of the vectors x and y ; Vf(x) is the gradient
vector of first order partial derivatives o f f evaluated at x ; V2f(x) is the
Hessian matrix of second order partial derivatives o f f evaluated at x ;
S u T means the union of the sets S and T; and Sn T is the intersection of
S and T.
1. Duality between unit cost and production functions
Assume we are given an N factor production function f, where y =f(x, ,
x2, ...,xN)=f(x) means y is the maximal amount of output which can be
produced during a given period of time using xi units of input i for i =
= 1,2, ..., N, where x = (x,, x,,
xN) is the vector of input levels. If the
...,
A set S in RN (Euclidean N space) is closed if 2'E S for n = 1,2, ..., lim x"= x0
implies x0 E S.
A set S in RN is convex if for every x E S, y E S and scalar 1 such that 0 51_< 1,
we have k + ( l - 1 ) Y E S.
A halfspace in R N is a set of'the form { x : a T x 5 k } , where aTx =
a,xl is the
inner product of the vectors a and x. See Fenchel(1953, pp. 48-50) or Rockafellar (1970,
pp. 95-99) for a treatment of Minkowski's theorem.
1g
j@
4
2
F
;*,
+-
Applications of duality theory
production function satisfies certain regularity conditions, then we may
calculate the producer's total minimum cost function C(y ;p l ,p 2 , ...,pN)=
= C(y; p), where p = (p, ,p,, ...,p,) is the vector of input prices, as the
solution to the following constrained minimization problem:
C(y;p)
i
= minx {pTx: f (x) 2 y).
In other words, the producer takes prices as given and attempts to minimize
the cost of producing a specified output level, y. In general, total cost C
depends on y, the chosen output level; p, the given vector of input prices;
and f, the given production function.
A production function f determines a cost function C through definition
(1). What is not as well known, is the converse, that a cost function satisfying
certain regularity conditions determines a production function; that is, there
is a duality between cost and production functions. Given one of these functions, under certain regularity conditions, the other can be uniquely determined, a result originally due to Shephard (1953) and Samuelson (1953-4).
The production function can in general be obtained from a cost function
satisfying the appropriate regularity conditions as the solution to the following constrained maximization problem:
-
f * ( ~ ) max, {y: C(y; p)
I
I
_< p T for
~
every p 2 ON),
(2)
where R = (Z,, Z,, ..., ZN)is a given vector of inputs and C is the given cost
function. Note that problem (2) has an infinite number of constraints.
A geometric interpretation of the maximization problem (2) can be obtained as follows: for every vector of factor prices p > ON,we can graph the
set of input combinations x such that pTx = C(y; p) for some fixed output
level y. Since C(y;p) is supposed to represent the minimum cost of producing
output y given prices p, it is reasonable to assume that the set L(y) =
= {x:f(x) 2 y) lies within the halfspace H(y; p) = { X : ~2~C(y;
X p)). In
fact, we may define the envelope production possibilities set L4(y) as the
intersection over all price vectors p > 0, of the halfspaces H(y;p); i.e.,
L*(y) = " p > ON H(y; p). Now choose j so that Z belongs to the boundary
of L* ( j ) ,and we have determined j =f(Z) using only the given cost function
C. Moreover, we will have C(7; p) = pTZ for some p > ON,where C(j;p) 5
S p T 2 for every p 2 ON. Notice that we have defined the set L"(y) as the
intersection of a family of halfspaces, where each halfspace was defined by an
isocost surface. In order for the envelope production possibilities setL* (y) t o
coincide with the "true" production possibilities set L(y) = {x:f(x) 2 y),
/
>
is,-
i,&-
W. E. Diewert
Mangasarian, O.L., 1969, Nonlinear Programming, New York, McGraw-Hill.
Rockafellar, R.T.. 1970, Convex Analysis, Princeton, New Jersey, Princeton University
Press.
Shephard, R.W., 1953, Cost and Production Functions, Princeton, New Jersey, Princeton
Universitv Press.
Shephard, R. W., 1970, Theory of Cost and Production Functions, Princeton, New Jersey,
Princeton University Press.
F P O X ~+,c 0&MQrsk+Q+;Ja
~
Eco*-;c~ % 1 x)&dB
M m - r ~BY LAWRENCE J. L A U ~
Stanford University
Applications of duality theory
[ g?
25.
4 @
I
{b "I:- <..
-{z
r$.
.-*
:
1
'
haD*Te+ri / ;
3
0. Introduction
F*(Y*) = sup { ( y , Y*) - F ( Y ) ) ,
Y
,
I
1. Some definitional, methodological and terminological issues
1 .O. Some preliminaries
Before taking up the issues, a few preliminary remarks about conventions and
notations used in this comment are in order.
First, the quantities of commodities are measured with positive signs
when they are net outputs and negative signs when they are net inputs.
* The author thanks Erwin Diewert, Arthur Goldberger, Dale Jorgenson and Daniel
McFadden for very useful discussions but retains full responsibility for any error contained
herein. Financial support from the National Science Foundation through Grant No. GS2874-A1 to the Institute of Mathematical Studies in the Social Sciences of Stanford
University is gratefully acknowledged.
Second, the set of production possibilities is assumed to be contained in
R"+', that is, there are (n+ 1) distinct commodities.
Third, the function which gives the value of the supremum of the negative
of the quantity of the (n 1)st commodity contained in the set of production
possibilities for given values of the quantities of the remaining n commodities
is defined as the production function and denoted F(y), where y is a vector
in Rn having as its elements the quantities of the 11 remaining commodities.
It is assumed that the (n + 1)st commodity is variable and freely disposable.'
The reader will note that this function reduces to the negative of the conventional production function in the one-output case. Other terminology for
F(y) includes the joint production function, the production possibility
frontier and the transformation function.' I shall refer to this function as the
production function.
Fourth, the productibn function F(jl) is an extended-real-valued function,
that is, it is permitted to take infinite values. F(j1) is said to be proper if it is
finite for some y and does not take the value minus infinity anywhere.
Fifth, the conjugate convex function of F(y), denoted F*(y*), is given by
+
i .!j
Professor Erwin Diewert has provided us with a very sweeping survey of the
applications of duality theory in the many different and seemingly unrelated
areas of economics. His survey serves to give us a flavor of what can be done
with duality theory. And the potential appears to be unbounded. He has
included in his survey, not only the results of published research, but also
many as yet unpublished original results of his own. We should thank
Erwin Diewert for this very excellent survey of a relatively new area of
research which will be useful for years to come.
In what follows I would like to put forth my own views on duality
theory, which, I hope, complement Diewerts' survey, and clarify some
issues. My comment can be divided ~ n t ofour sections: (1) Some definitional,
methodological and terminological issues; (2) Implications of duality for
applied econometrics; (3) More applications of duality theory; and (4)
Directions for related research.
177
I
where (,) denotes the inner product operation and they" may be interpreted
as normalized prices, that is, the prices of the commodities divided by the
price of the (n+ 1)st commodity. It may be referred to as the normalized
profit function because it can be identified with the profit function, a function
of (n+ 1) prices, if the price of the (n+ 1)st commodity is set equal to unity.
Other terminology for F* (y*) includes the Unit-Output-Price of UOP profit
f ~ n c t i o n I. ~shall refer to the F* (y*) as the normalized profit function.
Sixth, of special interest is the case in which the quantities of a subset of
the commodities are fixed. Adhering to the convention of always choosing
a variable and freely disposable commodity as the (11 + 1)st commodity, one
may partition a given y into two mutually exclusive subsets, y', which
consists of variable commodities, anti z, which consists of fixed commodities.
f r c c disposal of the (,I+I)st commodity is requircd for the one-to-one correspondence betwen a closed, convex set of production possibilities and a closed, convex function
F(y). If the free disposal assumption is droppcd, two I'unctions, F+(y) and F-(y) will be
required to characterize the convex set of production possibilities completely.
This terminology is uscd by Diewert (1973). However, i t is different from ;L related
concept by the same name used by Mcfitdden (19734.
This terminology is used in Lau (1969).
Applications of duality theory
The conjugate convex function of F(y', z), considered a function of y',
denoted F* (y'*, z), is given by
:I its modern form, this theory states that given F(y), a closed, proper and
f $
.
E
where the y'* may again be interpreted as normalized prices. It may be
referred to as the normalized restricted profit function: because it can be
identified with the restricted profit function if the price of the (n+ 1)st
commodity is set equal to unity. Other terminology for the restricted profit
function include gross profit function,' partial profit f ~ n c t i o n and
, ~ variable
profit function.' We note that in the case that z is a vector of outputs, and y'
is a vector of inputs, F* (yl*, z) is the negative of the normalized cost function; in the case that z is a vector of inputs, and y' is a vector of outputs,
F*(yl*, z ) is the normalized revenue function; in the case that the quantities
of all commodities are variable, the normalized restricted profit function is
the normalized profit function.
Seventh, the subgradient of a convex function F(y) at y, denoted y*, is
defined by the system of inequalities
F(x) 2 F(y)
+ (y*,x-y)
1. I. Alternative approaches to duality theory
There are as many alternative approaches to duality theory as there are
individuals working in the field of duality theory. These different approaches
may be approximately classified, at the risk of gross simplification, into three
groups.
The&~f,gfrup 2f approaches is based on the conjugacy correspondence
developed by ~enchel'(l949,1953) and extended by Rockafellar (1970a). In
'
This
This
This
This
terminology
terminology
terminology
terminology
I
!
follows McFadden (1973a).
is used by Gorrnan (1968).
is used by Lau (1969).
is used by Diewert (1973).
r
convex function, its conjugate dual, defined as
F*(Y*) = sup {(Y,y*) - F(y)}
Y
is also a closed proper convex fu~~ction.
Moreover, the dual of the dual,
defined as
is equal to F(y):
F * * ( ~ )= F(y).
Vx.
The set of all subgradients at y denoted aF(y) is referred to as the subdifferential of F(y) at y. If the subdifferential at y consists of only one element, it is
equal to the gradient of F Q at y, denoted VF(y). Given the subdifferential,
a closed, proper and convex function is determined up to a n additive
constant.
179
g
Hence there is a one-to-one correspoi~dencebetween F(y) and F*(y*).
Economically, this implies a one-to-one correspondence between the
production function and the normnlized profit function under the assumption
of closure, properness, and convexity. The precursor of this conjugacy
correspondence is of course the classical Legendre transformation. This
transformation was implicit in the pioneering work of Hotelling (1932) on
the normalized profit function. This line of reasoning was taken up by
Samuelson (1953) and Lau (1969) for the differentiable case, and by Jorgenson and Lau (1973) for the general case. One may add that with the additional
assumption of nonproducibility of the (n+ 1)st con~modity,that is, it must
be used as an input, the properties of F(y) and F*(y*) are completely
identical - they are both nonnegative and are equal to zero at the origin.'
Any function that can be used as a production function can also be used as a
normalized profit function, and vice versa, and the duality is completely
symmetric.
The second group of approaches is based on the symmetric duality
between gauge functions', or distance functions, or polar cones of convex
sets. Shephard (1953) was the pioneer of this group. He proved the duality
theorem between production and cost functions under the assumption of
differentiability employing the concept of distance functions. Other scholars
who have used similar approaches include Gorrnan (1968), McFadden
(1973a), Hanoch (1973), and Jacobsen ( 1 970, 1972), among others.
The third group of approaches is based 011the duality between the set of
prod"&i6n po%sibilitiesand its support function. The work of Uzawa (1964),
McFadden (1966) and Diewert (1971, 1973) may be classified amongst this
group.
Unskilled labor appears to be the natural choice for the
(?I+ ])st
commodity.
67.10 CONJUGATE CONVEX FUNCTIONALS
Let y be an arbitrary point in
e. Since
is (relatively) open, there is a
p > 1 such that By E c. Given E > 0,let 6 > 0 be such that llxll < 6 implies
If (x)l < E . Then for llz - yll < (1 - 8-')6, we have
z =y
for some x E
+ (1 - 8-')x
= p-'(py)
+ (1 - p-')x
c with llxll < 6. Thus z E C and
195
for each a E R. It follows that the set
T, = {x : x E C, f (x) I a)
is closed.
Now suppose {xi) is a sequence from C converging to x E C: Let
b = lim inf f (xi). If b = - co, then x E T, = To for each a E R which is
X,+X
impossible. Thus b > - co and x E Tb+£= Tb+£for all E > 0. In other
words, f (x) I lim inf f (xi) which proves that f is lower semicontinuous.
X,+X
Thus f is bounded above in the sphere llz - yll < (1 - 8-')a. It follows
that for sufficiently large r the point (r, y) is an interior point of [f, C];
hence, by Proposition 1,f is continuous at y. I
Figure 7.9 shows the graph of a convex functionalf defined on a disk C
in E2 that has closed [f,C] but is discontinuous (although lower semicontinuous) at a point x.
The proof of the following important corollary is left to the reader.
Corollary 1. A convexfunctional defined on a finite-dimensional convex set C
is continuous throughout
c.
Having established the simple relation between continuity and interior
points, we conclude this section by noting a property off which holds i j
[f, C] happens to be closed. As illustrated in Figure 7.8, closure of
[f,C] is related to the continuity properties off on the boundary of C.
Figure 7.9
7.10 Conjugate Convex Functionals
Figure 7.8 A nonclosed epigraph
Proposition 3.
If [f, C]
is closed, then f is lower semicontinuous on C.
Proof. The set {(a, x) E R x X : x E X) is obviously closed for each
a E R. Hence, if [f,C] is closed, so is
[f,C] n { ( a , ~ :)x E X) = { ( a , ~ )x: i C;f(x) 5 a)
A purely abstract approach to the theory of convex functionals, including
a study of the convex set Lf, C] as in the previous section, leads quite
naturally to an investigation of the dual representation of this set in terms
of closed hyperplanes. The concept of conjugate functionals plays a natural
and fundamental role in such a study. As an important consequence of this
investigation, we obtain a very general duality principle for optimization
problems which extends the earlier duality results for minimum norm
problems.
Definition. Let f be a convex functional defined on a convex set C in a
normed space X. The conjugate set C* is defined as
c * = {x*
E
x*:sup [(x, x*>
x EC
- f ( x ) ] < co)
$7.10
and the functional f * conjugate to f is defined on C * as
f *(x*) = sup [(x, x*) - f (~11.
CONJUGATE CONVEX FUNCTIONALS
197
The supremum on the right is achieved by some x since the problem is
finite dimensional. We h d , by differentiation, the solution
= IxrlP-' sgn xi
x 0C
Proposition 1. The conjugate set C* and the conjugate functional f * are
convex and [f *, C *] is a closed convex subset of R x X *.
Proof. For any x:, x;
SUP {(x, ux:
x 0C
E
X* and any u, 0 < u < 1, we have
+ (1 - u)x,*) - f (x)) = xsupC {uC(x, x:) - f (x)]
E
+ (1 - u)C(x, x;) - f (xlll
Or SUP C(x, xr> - f ( 4 1
x EC
+ (1 - a) XSUEPC C(x, x;> - f (x)l
from which it follows immediately that C* and f * are convex.
Next we prove that [f *, C*] is closed. Let {(st, x*)) be a convergent
sequence from [f *, C*] with (s,, x:) + (s, x*). We show now that
(s, x*) E [f *, C *]. For every i and every x E C, we have
sr 2 f *($) 2 (x, x:)
- f (x).
where l/p + l/q = 1.
Let us investigate the relation of the conjugate functional to separating
hyperplanes. On the space R x X, closed hyperplanes are represented by
(r,X) E IRxX
an equation of the form
((b
(x, x*)
s 2 (x, x*) - f (x)
E
C. Therefore,
(s,x%
RXK*
=k
-r =k
describe parallel closed hyperplanes in R x X. The number f *(x*) is the
supremum of the values of k for which the hyperplane intersects [f, C].
Thus the hyperplane (x, x*) - r =f*(x*) is a support hyperplane of
CI.
In the terminology of Section 5.13, f *(x*) is the support functional
h[(- 1, x*)} of the functional (- 1, x*) for the convex set [f, C]. The
special feature here is that we only consider functionals of the form
(- 1, x*) on R x X and thereby eliminate the need of carrying an extra
variable.
For the application to optimization problems, the most important geometric interpretation of the conjugate functional is that it measures vertical
distance to the support hyperplane. The hyperplane
cf,
s 2 sup C(x, x*) - f
XEC
from which it follows that x* E C* and s 2 f *(x*). )
We see that the conjugate functional defines a set [f *, C*] which is of
the same type as [f, C] ; therefore we write [f,
= Lf *, C*]. Note that
iff = 0, the conjugate functionalf * becomes the support functional of C.
a*
Example 1. Let X = C = En and define, for x = (x,, x, , . . . , x,), f (x) =
l/p
IxllP, 1 < p c a. Then for x* = (5,, 5,, . .. , c,),
x!=,
(s,fl>> = sr + (x, x*)
where s, k, and x* determine the hyperplane. Recalling that we agreed to
refer to the R axis as vertical, we say that a hyperplane is nonvertical if it
intersects the R axis at one and only one point. This is equivalent to the
requirement that the defining linear functional (s, x*) have s # 0. If attention is restricted to nonvertical hyperplanes, we may, without loss of
generality, consider only those linear functionals of the form (- 1, x*).
Any nonvertical closed hyperplane can then be obtained by appropriate
choice of x* and k.
To develop a geometric interpretation of the conjugate functional, note
that as k varies, the solutions (r, x) of the equation
Taking the limit as i -+ a , we obtain
for all x
,X ) ,
intersects the vertical axis (i.e., x =8) at (- f *(x*), 8). Thus, -f *(x*) is
the vertical height of the hyperplane above the origin. (See Figure 7.10.)
198
OPTIMIZATION OF FUNCTIONALS
7
57.1 1
199
CONJUGATE CONCAVE FUNCTIONALS
We prove the converse by contraposition. Let (r,, x,) q! [f, C]. Since
[f, C] is closed, there is a hyperplane separating (r,, x,) and [f,C].
Thus there exist x* E X*, s, and c such that
Figure 7.10 A conjugate convex functional
Another interpretation more clearly illuminates the duality between
for all (r, X) E [f, C]. It can be shown that, without loss of generality, this
hyperplane can be assumed to be nonvertical and hence s # 0 (see Problem 16). Furthermore, since r can be made arbitrarily large, we must have
s < 0. Thus we take s = - 1. Now it follows that (x, x*) - f (x) I c
for all x E C, which implies that (c, x*) E [f *, C*]. On the other
hand, c < (x, , x*) - r, implies (x, , x*) - c > r, , which implies that
(r0, xo) $ *[f*, C*l. I
[f, C] and [f*, C*] in terms of the dual representation of a convex set
7.11 Conjugate Concave Functionals
as a collection of points or as the intersection of half-spaces. Given the
point (s, x*) E R x X*, let us associate the half-space consisting of all
(r, x) E R x X satisfying
(x, x*) - r l s.
A development similar to that of the last section applies to concave functional~.It must be stressed, however, that we do not treat concave functional~by merely multiplying by - 1 and then applying the theory for
convex functionals. There is an additional sign change in part of the
definition. See Problem 15.
Given a concave functional g defined on a convex subset D of a vector
space, we define the set
Then the set [f *, C *] consists of those (nonvertical) half-spaces that contain the set [f, C]. Hence [f *, C*] is the dual representation of [f , C].
Beginning with an arbitrary convex functional cp defined on a convex
subset r of a dual space X*, we may, of course, define the conjugate of cp
in X** or, alternatively, following the standard pattern for duality relations (e.g., see Section 5.7), define the set * r in X as
The set [g, Dl is convex and all of the results on continuity, interior points,
etc., of Section 7.9 have direct extensions here.
Definition. Let g be a concave functional on the convex set D. The conjugate set D* is defined as
and the convex functional
*cp(x) = SUP C(x, x*) - cp(x*)I
x*
€
r
on * r . We then write *[p, r] = [*cp, *r]. With these definitions we have
the following characterization of the duality between a convex functional
and its conjugate.
Proposition 2. Let f be a convex functional on the convex set C in a normed
space X. If [f, C] is closed, then [f, C] = *[[f, CI*].
Proof. We show first that [f, C] c *[f*, C*] = *[[f, CI*]. Let
(r, x) E [f, C] ; then for all x* E C*, f *(x*) 2 (x, x*) -f (x). Hence, we
have r 2 f(x) 2 (x, x*) -f *(x*) for all x* E C*. Thus
r 2 sup [(x, x*)- f *(x*)l
x* € C*
and (r,x) ~ * [ f * ,C*].
[g, Dl = ((r, X) : x E D, r Ig(x)).
D* = (x*
E
X*: inf[(x, x*) - g(x)] > -a),
xeD
and the functional g* conjugate to g is defined as
g*(x*) = inf [(x, x*) - g(x)]:,Srp
xeD
xcD
-<
f-f~x)
X,
xc>l
We can readily verify that D* is convex and that g* is concave. We
write [g, Dl* = [g*, D*].
Since our notation does not completely distinguish between the development for convex and concave functionals, it is important to make clear
which is being employed in any given context. This is particularly true when
the original function is linear, since either definition of the conjugate
functional might be employed and, in general, they are not equal.