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 |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.01.2023
|
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| ISSN: | 0025-5610, 1436-4646 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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
$$x\in \mathbb {R}^n$$
x
∈
R
n
if and only if there exists a sum-of-squares (SOS) polynomial
$$\sigma (x)$$
σ
(
x
)
such that
$$\sigma P$$
σ
P
is a sum of sparse SOS matrices. Second, we show that setting
$$\sigma (x)=(x_1^2 + \cdots + x_n^2)^\nu $$
σ
(
x
)
=
(
x
1
2
+
⋯
+
x
n
2
)
ν
for some integer
$$\nu $$
ν
suffices if
P
is homogeneous and positive definite globally. Third, we prove that if
P
is positive definite on a compact semialgebraic set
$$\mathcal {K}=\{x:g_1(x)\ge 0,\ldots ,g_m(x)\ge 0\}$$
K
=
{
x
:
g
1
(
x
)
≥
0
,
…
,
g
m
(
x
)
≥
0
}
satisfying the Archimedean condition, then
$$P(x) = S_0(x) + g_1(x)S_1(x) + \cdots + g_m(x)S_m(x)$$
P
(
x
)
=
S
0
(
x
)
+
g
1
(
x
)
S
1
(
x
)
+
⋯
+
g
m
(
x
)
S
m
(
x
)
for matrices
$$S_i(x)$$
S
i
(
x
)
that are sums of sparse SOS matrices. Finally, if
$$\mathcal {K}$$
K
is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for
$$(x_1^2 + \cdots + x_n^2)^\nu P(x)$$
(
x
1
2
+
⋯
+
x
n
2
)
ν
P
(
x
)
with some integer
$$\nu \ge 0$$
ν
≥
0
when
P
and
$$g_1,\ldots ,g_m$$
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. 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. |
| Author | Zheng, Yang Fantuzzi, Giovanni |
| Author_xml | – sequence: 1 givenname: Yang orcidid: 0000-0003-1545-231X surname: Zheng fullname: Zheng, Yang email: zhengy@eng.ucsd.edu organization: Department of Electrical and Computer Engineering, University of California – sequence: 2 givenname: Giovanni orcidid: 0000-0002-0808-0944 surname: Fantuzzi fullname: Fantuzzi, Giovanni organization: Department of Aeronautics, Imperial College London |
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| CitedBy_id | crossref_primary_10_1287_moor_2023_0361 crossref_primary_10_1007_s10898_020_00973_1 crossref_primary_10_1109_TAC_2025_3545766 crossref_primary_10_1007_s10107_023_01993_x crossref_primary_10_1137_24M1675461 crossref_primary_10_1137_23M1626529 |
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| Keywords | Polynomial matrix inequalities 11E25 Polynomial optimization 90C25 90C23 90C06 49M27 Chordal decomposition 12D15 11E76 |
| Language | English |
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| Title | Sum-of-squares chordal decomposition of polynomial matrix inequalities |
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