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The next step consists of solving (9.20) via convex optimization. To this end, let
us introduce the polynomials
z ) h i , j , k ( a
2
b r ( a
,
z )=
γ
a
u i , j , k ( a
,
,
z )
( i
,
j
,
k )
∈I
(9.21)
z ) h i , j , k ( a
2 ) 2
2
b t ( a
,
z )=(1 +
a
γ
z
u i , j , k ( a
,
,
z )
( i
,
j
,
k )
∈I
where u i , j , k ( a
is an auxiliary scalar to be
determined. Let us exploit the SMR of polynomials described in Section 9.2.3. Let
v b ( a
,
z )
R
are auxiliary polynomials and
γ R
,
z ) be a vector containing any base for the polynomials b r ( a
,
z ) and b t ( a
,
z ),and
let v u ( a
,
z ) be a similar vector for the polynomials u i , j , k ( a
,
z ). Then, the polynomials
b r ( a
,
z )
,
b t ( a
,
z ) and u i , j , k ( a
,
z ) can be expressed as
z ) T B r v b ( a
b r ( a
,
z )= v b ( a
,
,
z )
z ) T B t v b ( a
b t ( a
,
z )= v b ( a
,
,
z )
(9.22)
z ) T U i , j , k v u ( a
u i , j , k ( a
,
z )= v u ( a
,
,
z )
where B r , B t and U i , j , k are any symmetric matrices of suitable dimensions satisfying
(9.22). Let L (
α
) be any linear parametrization of the linear set
= L = L T
z
z ) T Lv b ( a
L
: v b ( a
,
,
z )=0
a
,
where
α
is a free vector, and let us define the optimization problems
s.t. B r + L (
α
)
0
γ r =
γ , α , U i , j , k γ
inf
U i , j , k
0
( i
,
j
,
k )
∈I
s.t. B t + L (
(9.23)
α
)
0
γ t =
γ , α , U i , j , k γ
inf
U i , j , k
0
( i
,
j
,
k )
∈I .
These optimization problems are convex since the cost functions are linear and the
feasible sets are convex being the feasible sets of LMIs. In particular, these prob-
lems belong to the class of eigenvalue problems (EVPs), also known as semidefinite
programming [2].
Let us observe that the constraints in (9.23) ensure that
b r ( a
,
z )
0
b t ( a
,
z )
0
a
,
z
u i , j , k ( a
,
z )
0
from which one obtains
2
γ r
a
2
a
,
z : h i , j , k ( a
,
z )
0
( i
,
j
,
k )
∈I .
z
γ t
2 ) 2
(1 +
a
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