POS3POLY—a MATLAB preprocessor for optimization with positive polynomials

Abstract

Positive polynomials, relaxed to sum-of-squares in the multivariate case, are a powerful instrument having applications in signal processing, control and other engineering fields. Hence, appeared the need of a library which can work with positive polynomials as variables in a convex optimization problem. We present here the POS3POLY library, which transforms polynomial positivity into positive semidefinite constraints, thus enabling the user to solve such problems without the need of knowing the parameterization for each type of polynomial. POS3POLY is able to handle three types of polynomials: trigonometric, real and hybrid. The positivity of the polynomials can be global or only on a semialgebraic domain. POS3POLY allows also to define Bounded Real Lemma constraints. The library is written in MATLAB and uses SeDuMi for solving the convex optimization problems. POS3POLY can also work inside CVX. To show the usage of our library we give several examples of 2-D FIR filter design.

Appendix: Parameterizations

1.1 A.1 Sum-of-squares

To illustrate the parameterization behind a sum-of-squares polynomial we give the next result for scalar trigonometric polynomials.

Theorem 1

A polynomial (4) is sum-of-squares if and only if there exists a positive semidefinite matrix Q∈ℂ^N×N such that

$$ r_{\mathbf{k}}=\mathbf{Tr}[\boldsymbol{\Theta}_{k_d}\otimes\cdots \otimes\boldsymbol{\Theta}_{k_1}\cdot\mathbf{Q}], \quad\mathbf{k}\in\mathcal{H}_d,$$

(41)

where Θ _k is the elementary Toeplitz matrix with ones on the k-th diagonal and zeros elsewhere and $N=\prod_{i=1}^{d}(n_{i}+1)$. The matrix Q is called a Gram (Parrilo 2003; Choi et al. 1995; McLean and Woerdeman 2001) matrix associated with the polynomial (4).

Hence, in order to create a scalar trigonometric sum-of-squares polynomial, POS3POLY implements the equalities from (41).

All positive trigonometric polynomials are sum-of-squares. However, the square polynomials appearing in the sum may have a degree that is higher than that of the positive polynomial, hence they would be parameterized with a matrix Q whose (unknown) size would be larger than N×N. In POS3POLY, the size that corresponds to the degree of the positive polynomial is used, hence we actually employ a relaxation.

For scalar real polynomials the parameterization is the following.

Theorem 2

A polynomial (10) with even degree n is sum-of-squares if and only if there exists a positive semidefinite matrix Q∈ℝ^N×N such that

$$ p_{\mathbf{k}} = \mathbf{Tr}[\boldsymbol{\Upsilon}_{k_d} \otimes\cdots \otimes\boldsymbol{\Upsilon}_{k_1}\cdot\mathbf{Q}],\quad\mathbf{0} \le\mathbf{k}\le{\mathbf{n}},$$

(42)

where ϒ _k is the elementary Hankel matrix with ones on the k-th antidiagonal and zeros elsewhere and $N=\prod_{i=1}^{d} (n_{i}/2+1)$.

Multivariate positive real polynomials are not necessarily sum-of-squares. Hence, using (42) for their parameterization is a relaxation.

To parameterize positivity on domains, for trigonometric or real polynomials, relation (24) is implemented using the above parameterizations for each sum-of-squares polynomial. This is again a relaxation, since in POS3POLY the parameterizations correspond to the minimal degrees of the sum-of-squares that make (24) possible, while (24) holds in general only if no restrictions are posed on the degrees of the sum-of-squares, which can be arbitrarily large.

1.2 A.2 Bounded real lemma

A BRL type constraint for scalar trigonometric polynomials is implemented in POS3POLY using the following result.

Theorem 3

Let H(z) and A(z) be causal polynomials. Denoting R(z)=A(z)A ^∗(z ⁻¹), the inequality

$$ |H(\boldsymbol{\omega})|\leq|A(\boldsymbol{\omega})|$$

is satisfied, if and only if there exists a matrix Q⪰0, such that the relations (41) and hold.