AbstractWe 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.
Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings
