Sum-of-squares chordal decomposition of polynomial matrix inequalities
We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert–Artin, Reznick, Putinar, and Putinar–Vasilescu Positivstellensätze. First, we establish that a polynomial matrix P ( x ) with chordal sparsity is p...
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| Vydané v: | Mathematical programming Ročník 197; číslo 1; s. 71 - 108 |
|---|---|
| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.01.2023
|
| Predmet: | |
| ISSN: | 0025-5610, 1436-4646 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert–Artin, Reznick, Putinar, and Putinar–Vasilescu Positivstellensätze. First, we establish that a polynomial matrix
P
(
x
) with chordal sparsity is positive semidefinite for all
x
∈
R
n
if and only if there exists a sum-of-squares (SOS) polynomial
σ
(
x
)
such that
σ
P
is a sum of sparse SOS matrices. Second, we show that setting
σ
(
x
)
=
(
x
1
2
+
⋯
+
x
n
2
)
ν
for some integer
ν
suffices if
P
is homogeneous and positive definite globally. Third, we prove that if
P
is positive definite on a compact semialgebraic set
K
=
{
x
:
g
1
(
x
)
≥
0
,
…
,
g
m
(
x
)
≥
0
}
satisfying the Archimedean condition, then
P
(
x
)
=
S
0
(
x
)
+
g
1
(
x
)
S
1
(
x
)
+
⋯
+
g
m
(
x
)
S
m
(
x
)
for matrices
S
i
(
x
)
that are sums of sparse SOS matrices. Finally, if
K
is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for
(
x
1
2
+
⋯
+
x
n
2
)
ν
P
(
x
)
with some integer
ν
≥
0
when
P
and
g
1
,
…
,
g
m
are homogeneous of even degree. Using these results, we find sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables, and we construct new convergent hierarchies of sparsity-exploiting SOS reformulations for convex optimization problems with large and sparse polynomial matrix inequalities. Numerical examples demonstrate that these hierarchies can have a significantly lower computational complexity than traditional ones. |
|---|---|
| ISSN: | 0025-5610 1436-4646 |
| DOI: | 10.1007/s10107-021-01728-w |