Positivstellensätze for polynomial matrices

Dinh, Trung Hoa; Ho, Minh Toan; Le, Cong Trinh

doi:10.1007/s11117-021-00816-7

Cited by 1 publication

(3 citation statements)

References 21 publications

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“…This result, proven in Sect. 6.4, recovers Theorem 4 in [9] when P is dense. If P is not homogeneous, we find the following version of the Putinar-Vasilescu Positivstellensätze [37] for sparse polynomial matrices, which is a sparsity-exploiting formulation of a recent result for general (dense) matrices [9, Corollary 3].…”

Section: Example 25supporting

confidence: 72%

“…We combine the argument given in [9] for general (dense) polynomial matrices with Theorem 2.4 and the following auxiliary result, proven in Appendix D. Lemma 6.1 Let S(x) be an SOS polynomial matrix satisfying S(x) = S(−x). For any real number r ≥ 0 and any integer ω such that 2ω ≥ deg(S), the matrix x 2ω S(r x −1 x) is polynomial of degree 2ω, homogeneous, and SOS.…”

Section: Proof Of Theorem 25mentioning

confidence: 99%

“…, m} and edges E ⊆ V × V such that Scherer and Hol [42] Theorem 2. 4 Putinar-Vasilescu Putinar and Vasilescu [37] Dinh et al [9] Theorem 2.5 P i j (x) = P ji (x) ≡ 0 when i = j and (i, j) / ∈ E. Motivated by chordal decomposition techniques for semidefinite programming [11,30,45,46,55], we ask whether the computational complexity of (1.3) can be lowered by decomposing the matrices S 0 , . .…”

Section: Introductionmentioning

confidence: 99%

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Sum-of-squares chordal decomposition of polynomial matrix inequalities

Zheng

Fantuzzi

2021

Math. Program.

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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.

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Section: Example 25supporting

confidence: 72%

Section: Proof Of Theorem 25mentioning

confidence: 99%