The level curves of 'pk trk are shown in gure 11.4. A subgradient g @'pk trk( )
2
is given by
2
3
Z
1
sgn(h(t))h1(t)dt
g =
sg(H1)
6
7
sg
0
(H
6
7
2) =
Z
6
7
1
4
sgn(h(t))h2(t)dt 5
0
where h is the impulse response of H0 + H1 + H2, h1 is the impulse response of
H1, and h2 is the impulse response of H2 (see section 13.4.6).
Consider the point = 1, = 0, where 'pk trk(1 0) = 0:837. A subgradient at
this point is
g =
0:168
0:309 :
(13.6)
;
In gure 13.6 the level curve 'pk trk( ) = 0:837 is shown, together with the
subgradient (13.6) at the point 1 0]T. As expected, the subgradient determines a
half-space that contains the convex set
]T 'pk trk(
) 'pk trk(1 0) = 0:837 :
2
0 837
:
1:5
1
0:5
x
q
?
0
g
;0:5
;1
;1
;0:5
0
0:5
1
1:5
2
The level curve pk trk(
) = 0 837 is shown, together with
Figure
13.6
'
:
the subgradient (13.6) at the point 1 0] .
T
NOTES AND REFERENCES
309
Notes and References
Convex Analysis
Rockafellar's book
] covers convex analysis in detail. Other texts covering this
R
oc70
material are Stoer and Witzgall
], Barbu and Precupanu
], Aubin and Vin-
SW70
BP78
ter
], and Demyanov and Vasilev
]. The last three consider the innite-
A
V82
D
V85
dimensional (Banach space) case, carefully distinguishing between the many important dif-
ferent types of continuity, compactness, and so on, which we have not considered. Daniel's