~ )
( )
)
H
H
;
;
(1;+
( ~)
H
;
(
;
H
= 0
= 1
;
;
;
;
A functional is ane if for every pair of transfer matrices
Figure
6.2
H
and ~ the graph of ( + (1
) ~) versus is a straight line passing
H
H
;
H
through the points (0 ( ~)), (1 ( )).
H
H
~ ))
( )
H
H
;
;
(1;+
( ~)
H
;
(
;
H
= 0
= 1
@
;
@
R
;
A functional is convex if for every pair of transfer matrices
Figure
6.3
and ~ the graph of along the line
+ (1
) ~ lies on or below a
H
H
H
;
H
straight line through the points (0 ( ~)), (1 ( )), i.e., in the shaded
H
H
region.
6.2 AFFINE AND CONVEX SETS AND FUNCTIONALS
131
Ane implies convex. If a set or functional is a ne, then it is convex: being
a ne is a stronger condition than convex. If the functionals and
are
;
both convex, then is a ne.
Intersections. Intersections of a ne or convex sets are a ne or convex, re-
spectively.
Weighted-sum Functional. If the functionals 1 ...
are convex, and 1
L
0, ...,
0, then the weighted-sum functional
L
wt sum( ) = 1 1( ) +
+
( )
H
H
H
L
L
is convex (see section 3.6.1).
Weighted-max Functional. If the functionals 1 ...
are convex, and 1
L
0, ...,
0, then the weighted-max functional
L
wt max( ) = max 1 1( ) ...
( )
H
f
H
H
g
L
L
is convex (see section 3.6.3).
The last two properties can be generalized to the integral of a family of convex
functionals and the maximum of an in nite family of convex functionals. Suppose
that for each
( is an arbitrary index set), the functional is convex. Then
2
I
I
the functional
( ) = sup
( )
H
f
H
j
2
I
g
is convex.
We now describe some of the relations between sets and functionals that are
convex or a ne. A functional equality speci cation formed from an a ne functional
de nes an a ne set: if is a ne and
, then
2
R
( ) =
fH
j
H
g
is a ne. Similarly, if is convex and
, then the functional inequality speci -
2
R
cation
( )
fH
j
H
g
called a sub-level set of , is convex.
The converse is not true, however: there are functionals that are not convex,
but every sub-level set is convex. Such functionals are our next topic.
132