Polynomial SOS

This article is about representing polynomial as sum of squares of polynomials. For representing polynomial as a sum of squares of rational functions, see Hilbert's seventeenth problem. For the sum of squares of consecutive integers, see square pyramidal number. For representing an integer as a sum of squares of integers, see Lagrange's four-square theorem.

In mathematics, a form (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only if there exist forms $g_1(x),\ldots,g_k(x)$ of degree m such that

h(x)=\sum_{i=1}^k g_i(x)^2 .

Explicit sufficient conditions for a form to be SOS have been found.^[1] However every real nonnegative form can be approximated as closely as desired (in the $l_1$ -norm of its coefficient vector) by a sequence of forms $\{f_\epsilon\}$ that are SOS.^[2]

Square matricial representation (SMR)

To establish whether a form h(x) is SOS amounts to solving a convex optimization problem. Indeed, any h(x) can be written as

h(x)=x^{\{m\}'}\left(H+L(\alpha)\right)x^{\{m\}}

where $x^{\{m\}}$ is a vector containing a base for the forms of degree m in x (such as all monomials of degree m in x), the prime ′ denotes the transpose, H is any symmetric matrix satisfying

h(x)=x^{\left\{m\right\}'}Hx^{\{m\}}

and $L(\alpha)$ is a linear parameterization of the linear space

\mathcal{L}=\left\{L=L':~x^{\{m\}'} L x^{\{m\}}=0\right\}.

The dimension of the vector $x^{\{m\}}$ is given by

\sigma(n,m)=\binom{n+m-1}{m}

whereas the dimension of the vector $\alpha$ is given by

\omega(n,2m)=\frac{1}{2}\sigma(n,m)\left(1+\sigma(n,m)\right)-\sigma(n,2m).

Then, h(x) is SOS if and only if there exists a vector $\alpha$ such that

H + L(\alpha) \ge 0,

meaning that the matrix $H + L(\alpha)$ is positive-semidefinite. This is a linear matrix inequality (LMI) feasibility test, which is a convex optimization problem. The expression $h(x)=x^{\{m\}'}\left(H+L(\alpha)\right)x^{\{m\}}$ was introduced in [1] with the name square matricial representation (SMR) in order to establish whether a form is SOS via an LMI. This representation is also known as Gram matrix (see [2] and references therein).

Examples

Consider the form of degree 4 in two variables $h(x)=x_1^4-x_1^2x_2^2+x_2^4$ . We have

m=2,~x^{\{m\}}=\left(\begin{array}{c}x_1^2\\x_1x_2\\x_2^2\end{array}\right), ~H+L(\alpha)=\left(\begin{array}{ccc} 1&0&-\alpha_1\\0&-1+2\alpha_1&0\\-\alpha_1&0&1 \end{array}\right).

Since there exists α such that

H+L(\alpha)\ge 0

, namely

\alpha=1

, it follows that h(x) is SOS.

Consider the form of degree 4 in three variables $h(x)=2x_1^4-2.5x_1^3x_2+x_1^2x_2x_3-2x_1x_3^3+5x_2^4+x_3^4$ . We have

m=2,~x^{\{m\}}=\left(\begin{array}{c}x_1^2\\x_1x_2\\x_1x_3\\x_2^2\\x_2x_3\\x_3^2\end{array}\right), ~H+L(\alpha)=\left(\begin{array}{cccccc} 2&-1.25&0&-\alpha_1&-\alpha_2&-\alpha_3\\ -1.25&2\alpha_1&0.5+\alpha_2&0&-\alpha_4&-\alpha_5\\ 0&0.5+\alpha_2&2\alpha_3&\alpha_4&\alpha_5&-1\\ -\alpha_1&0&\alpha_4&5&0&-\alpha_6\\ -\alpha_2&-\alpha_4&\alpha_5&0&2\alpha_6&0\\ -\alpha_3&-\alpha_5&-1&-\alpha_6&0&1 \end{array}\right).

Since

H+L(\alpha)\ge 0

for

\alpha=(1.18,-0.43,0.73,1.13,-0.37,0.57)

, it follows that h(x) is SOS.

Matrix SOS

A matrix form F(x) (i.e., a matrix whose entries are forms) of dimension r and degree 2m in the real n-dimensional vector x is SOS if and only if there exist matrix forms $G_1(x),\ldots,G_k(x)$ of degree m such that

F(x)=\sum_{i=1}^k G_i(x)'G_i(x) .

Matrix SMR

To establish whether a matrix form F(x) is SOS amounts to solving a convex optimization problem. Indeed, similarly to the scalar case any F(x) can be written according to the SMR as

F(x)=\left(x^{\{m\}}\otimes I_r\right)'\left(H+L(\alpha)\right)\left(x^{\{m\}}\otimes I_r\right)

where $\otimes$ is the Kronecker product of matrices, H is any symmetric matrix satisfying

F(x)=\left(x^{\{m\}}\otimes I_r\right)'H\left(x^{\{m\}}\otimes I_r\right)

and $L(\alpha)$ is a linear parameterization of the linear space

\mathcal{L}=\left\{L=L':~\left(x^{\{m\}}\otimes I_r\right)'L\left(x^{\{m\}}\otimes I_r\right)=0\right\}.

The dimension of the vector $\alpha$ is given by

\omega(n,2m,r)=\frac{1}{2}r\left(\sigma(n,m)\left(r\sigma(n,m)+1\right)-(r+1)\sigma(n,2m)\right).

Then, F(x) is SOS if and only if there exists a vector $\alpha$ such that the following LMI holds:

H+L(\alpha) \ge 0.

The expression $F(x)=\left(x^{\{m\}}\otimes I_r\right)'\left(H+L(\alpha)\right)\left(x^{\{m\}}\otimes I_r\right)$ was introduced in [3] in order to establish whether a matrix form is SOS via an LMI.

References

[1] G. Chesi, A. Tesi, A. Vicino, and R. Genesio, On convexification of some minimum distance problems, 5th European Control Conference, Karlsruhe (Germany), 1999.

[2] M. Choi, T. Lam, and B. Reznick, Sums of squares of real polynomials, in Proc. of Symposia in Pure Mathematics, 1995.

[3] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, Robust stability for polytopic systems via polynomially parameter-dependent Lyapunov functions, in 42nd IEEE Conference on Decision and Control, Maui (Hawaii), 2003.

Notes

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