The dual function can be used to determine speci cations that are not achiev-
able, and therefore represent a limit of performance (recall the discussion in sec-
tion 1.4.1). Whenever
0, meaning
L
+, and
L
, we have
2
R
a
2
R
( )
=
a
...
a
is unachievable
(6.9)
T
hard
1
L
>
a
)
D
^
D
^
^
D
:
1
L
To establish (6.9), suppose that corresponds to an achievable speci cation, i.e.,
a
hard
a
...
a
is achievable. Then there is some transfer matrix ~ that
1
L
D
^
D
^
^
D
H
1
L
satis es hard and ( ~)
for each . Therefore whenever
0,
D
H
a
i
i
i
( ) = min 1 1( ) + +
( )
satis es hard
f
H
H
j
H
D
g
L
L
1 1( ~ ) +
+
( ~)
H
H
L
L
1 1 +
+
a
a
L
L
= T
a
:
The implication (6.9) follows.
achievable
2
q
a
x
D
( )
T
>
a
unachievable
1
a
The shaded region corresponds to specications that satisfy
Figure
6.7
( )
, and hence by (6.9) are unachievable. The specication
T
x is
>
a
D
unachievable, but cannot be proven unachievable by (6.9), for any choice of
. See also gure 3.8.
The speci cations that (6.9) establishes are unachievable have the simple geo-
metric interpretation shown in gure 6.7, which suggests that when the region of
achievable speci cations is convex, (6.9) rules out all unachievable speci cations as
varies over all nonnegative weight vectors. This intuition is correct, except for a
technical condition. More precisely, when hard and each objective functional is
D
i
6.6 CONVEXITY AND DUALITY
141
convex, and the region of achievable speci cations in performance space is closed,
then we have:
the speci cation corresponding to is achievable
a
(6.10)
(
)
there is no
0 with ( )
T
>
a
:
The equivalence (6.10) is called the convex duality principle, or a theorem of
alternatives, since it states that exactly one of two alternative assertions must hold.
The geometrical interpretation of the duality principle can be seen from gure 6.8:
(6.10) asserts that every speci cation is either achievable (i.e., lies in the shaded
region to the upper right), or can be shown unachievable using (6.9) (i.e., lies in
the shaded region to the lower left) for some choice of .
achievable
2
a
( )
T
>
a
unachievable
1
a
Unlike gure 6.7, in this gure the region of achievable spec-
Figure
6.8
ications is convex, so, by varying , (6.9) can be made to rule out every
specication that is unachievable.
The convex duality principle (6.10) is often expressed in terms of pairs of con-
strained optimization problems, e.g., minimizing the functional 1 subject to the
hard constraint and functional inequality speci cations for the remaining function-
als 2 ... . We de ne
L
o
pri = min n 1( )
satis es hard a ... a
(6.11)
2
L
H
H
D
^
D
^
^
D
2
L
dual = max ( )
2 2
0 1 = 1
(6.12)
f
;
a
;
;
a
j
g
:
L
L
The optimization problem on the right-hand side of (6.11) is called the primal
problem the right-hand side of (6.12) is called a corresponding dual problem. The
142