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Theorem 10.4 Let x be a feasible solution of ( OMPF ) then there exists a solution
.x; u ;v; w / for ( 10.13 ) - ( 10.24 ) such that their objective values are equal. Con-
versely, if .x; u ;v; w / is a feasible solution for ( 10.13 ) - ( 10.24 ) then there ex is ts
a solution .x/ for ( OMPF ) having the same objective valu e. In particular D .
Moreover, if K
dCm 2
d satisfies Archimedean property then K
Cn.dC1/ also
R
R
satisfies Archimedean property.
The interested reader is referred to Blanco et al. ( 2013 , Theorem 4) for a detailed
proof.
Now, we can prove a convergence result that allows us to solve, up to any degree
of accuracy, the above class of problems. In order to proceed further we need to
introduce some additional material related to the Theory of Moments, Lasserre
( 2009 ).
Recall that by
Œx we denote the ring of real polynomials in the variables x D
.x 1 ;:::;x d /,ford 2
R
N
(d 1), and by
R
Œx r
R
Œx the space of polynomials
of degree at most r 2
N
(here
N
denotes the set of non-negative integers). We also
Df x Ǜ W Ǜ 2
d g a canonical basis of monomials for
denote by
B
N
R
Œx,where
W P iD1 Ǜ i r g is a
x Ǜ D x Ǜ 1 x Ǜ d ,foranyǛ 2
d . Note that
B r Df x Ǜ 2
N
B
basis for
Œx r .
For any sequence indexed in the canonical monomial basis
R
B
, y D .y Ǜ / Ǜ2 N
d
be the linear functional defined, for any f D P Ǜ2 N
d f Ǜ x Ǜ 2
R
,letL y W
R
Œx !
R
Œx,asL y .f / WD P Ǜ2 N
R
d f Ǜ y Ǜ .
The moment matrix M r . y / of order r associated with y , has its rows and columns
indexed by .x Ǜ / and M r . y /.Ǜ;LJ/ WD L y .x ǛCLJ / D y ǛCLJ ,for j Ǜ j ; j LJ j r (here
j a j stands for the sum of the coordinates of a 2
d ).
N
Œx. D P 2 N
d g x ), the localizing matrix M r .g y / of order r associ-
ated with y and g, has its rows and columns indexed by .x Ǜ / and M r .g y /.Ǜ;LJ/ WD
For g 2
R
L y .x ǛCLJ g.x// D P g y CǛCLJ ,for j Ǜ j ; j LJ j r.
Let y D .y Ǜ / be a real sequence indexed in the monomial basis .x LJ u v ı w / of
m 2
d
n
nd
R
).
Let h 0 .x; u ;v; w / WD p .x; u ;v; w /, and denote j WD d .degg j /=2 e and j WD
d .degh j /=2 e ,where f g 1 ;:::;g ` g ,and f h 1 ;:: :; h 3mCm 2 Cn.dC3/ g are, respectively,
the polynomial constraints that define K and K n K in ( 10.13 )-( 10.24 ). For r
r 0 WD max f max kD1;:::;` k ; max jD0;:::;3mCm 2 Cn.dC3/ j g , we introduce the hierarchy
of semidefinite programs:
Œx; u ;v; w (with Ǜ D .LJ;;ı;/ 2
N
N
N
N
minimize y L y .p /
subject to
M r . y / 0;
M r k .g k ; y / 0; k D 1;:::;`;
M r j .h j ; y / 0; j D 1;:::;3m C m 2 C n.d C 3/;
( Q r )
with optimal value denoted min Q r .
 
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