Polynomial SOS

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 ${\displaystyle g_{1}(x),\ldots ,g_{k}(x)}$ of degree m such that $${\displaystyle h(x)=\sum _{i=1}^{k}g_{i}(x)^{2}.}$$

Every form that is SOS is also a positive polynomial, although the converse is not always true in general. In the special cases of n = 2 and 2m = 2, or n = 3 and 2m = 4, Hilbert proved that a form is SOS if and only if it is positive.¹ The same is also true for the analogous problem with positive symmetric forms.²³

Although not every form is SOS, there are efficiently testable sufficient conditions for a form to be SOS.⁴⁵ Moreover, every real nonnegative form can be approximated as closely as desired (in the ${\displaystyle l_{1}}$-norm of its coefficient vector) by a sequence of forms ${\displaystyle \{f_{\epsilon }\}}$ that are SOS.⁶

To establish whether a form h(x) is SOS amounts to solving a convex optimization problem. Indeed, any h(x) can be written as $${\displaystyle h(x)=(x^{\{m\}})^{T}\left(H+L(\alpha )\right)x^{\{m\}}}$$ where ${\displaystyle x^{\{m\}}}$ is a vector containing a basis for the space of forms of degree m in x (such as all monomials of degree m), H is any symmetric matrix satisfying ${\displaystyle h(x)=(x^{\left\{m\right\}})^{T}Hx^{\{m\}}}$, and ${\displaystyle L(\alpha )}$ is a linear parameterization of the linear subspace $${\displaystyle {\mathcal {L}}=\left\{L=L':~x^{\{m\}'}Lx^{\{m\}}=0\right\}.}$$

The dimension of the vector ${\displaystyle x^{\{m\}}}$ is given by $${\displaystyle \sigma (n,m)={\binom {n+m-1}{m}},}$$ whereas the dimension of the vector ${\displaystyle \alpha }$ is given by $${\displaystyle \omega (n,2m)={\frac {1}{2}}\sigma (n,m)\left(1+\sigma (n,m)\right)-\sigma (n,2m).}$$

The polynomial h(x) is SOS if and only if there exists a vector ${\displaystyle \alpha }$ such that ${\displaystyle H+L(\alpha )}$ is a positive-semidefinite matrix. This is a linear matrix inequality (LMI), and the existence of ${\displaystyle \alpha }$ is a convex feasibility problem.

The expression ${\displaystyle h(x)=x^{\{m\}'}\left(H+L(\alpha )\right)x^{\{m\}}}$ was introduced with the name square matricial representation (SMR) in order to establish whether a form is SOS via an LMI.⁷ The matrix ${\displaystyle H+L(\alpha )}$ is also known as a Gram matrix.⁸

• Consider ${\displaystyle m=2}$ and the form ${\displaystyle h(x)=x_{1}^{4}-x_{1}^{2}x_{2}^{2}+x_{2}^{4}}$ of degree 4 in two variables. We have $${\displaystyle x^{\{m\}}={\begin{pmatrix}x_{1}^{2}\\x_{1}x_{2}\\x_{2}^{2}\end{pmatrix}}\!,~H+L(\alpha )={\begin{pmatrix}1&0&-\alpha _{1}\\0&-1+2\alpha _{1}&0\\-\alpha _{1}&0&1\end{pmatrix}}\!.}$$ Since there exists α such that ${\displaystyle H+L(\alpha )\geq 0}$, namely ${\displaystyle \alpha =1}$, it follows that h(x) is SOS.
• Consider ${\displaystyle m=2}$ and the form ${\displaystyle h(x)=2x_{1}^{4}-5x_{1}^{3}x_{2}/2+x_{1}^{2}x_{2}x_{3}-2x_{1}x_{3}^{3}+5x_{2}^{4}+x_{3}^{4}}$ of degree 4 in three variables. We have $${\displaystyle x^{\{m\}}={\begin{pmatrix}x_{1}^{2}\\x_{1}x_{2}\\x_{1}x_{3}\\x_{2}^{2}\\x_{2}x_{3}\\x_{3}^{2}\end{pmatrix}},~H+L(\alpha )={\begin{pmatrix}2&-5/4&0&-\alpha _{1}&-\alpha _{2}&-\alpha _{3}\\-5/4&2\alpha _{1}&1/2+\alpha _{2}&0&-\alpha _{4}&-\alpha _{5}\\0&1/2+\alpha _{2}&2\alpha _{3}&\alpha _{4}&\alpha _{5}&-1\\-\alpha _{1}&0&\alpha _{4}&5&0&-\alpha _{6}\\-\alpha _{2}&-\alpha _{4}&\alpha _{5}&0&2\alpha _{6}&0\\-\alpha _{3}&-\alpha _{5}&-1&-\alpha _{6}&0&1\end{pmatrix}}.}$$ Since ${\displaystyle H+L(\alpha )\geq 0}$ for ${\displaystyle \alpha =(1.18,-0.43,0.73,1.13,-0.37,0.57)}$, it follows that h(x) is 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 ${\displaystyle G_{1}(x),\ldots ,G_{k}(x)}$ of degree m such that $${\displaystyle F(x)=\sum _{i=1}^{k}G_{i}(x)^{T}G_{i}(x).}$$

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 $${\displaystyle F(x)=\left(x^{\{m\}}\otimes I_{r}\right)^{T}\left(H+L(\alpha )\right)\left(x^{\{m\}}\otimes I_{r}\right)}$$ where ${\displaystyle \otimes }$ is the Kronecker product of matrices, H is any symmetric matrix satisfying $${\displaystyle F(x)=\left(x^{\{m\}}\otimes I_{r}\right)^{T}H\left(x^{\{m\}}\otimes I_{r}\right)}$$ and ${\displaystyle L(\alpha )}$ is a linear parameterization of the linear space $${\displaystyle {\mathcal {L}}=\left\{L=L^{T}:~\left(x^{\{m\}}\otimes I_{r}\right)^{T}L\left(x^{\{m\}}\otimes I_{r}\right)=0\right\}.}$$

The dimension of the vector ${\displaystyle \alpha }$ is given by $${\displaystyle \omega (n,2m,r)={\frac {1}{2}}r\left(\sigma (n,m)\left(r\sigma (n,m)+1\right)-(r+1)\sigma (n,2m)\right).}$$

Then, F(x) is SOS if and only if there exists a vector ${\displaystyle \alpha }$ such that the following LMI holds: $${\displaystyle H+L(\alpha )\geq 0.}$$

The expression ${\displaystyle F(x)=\left(x^{\{m\}}\otimes I_{r}\right)^{T}\left(H+L(\alpha )\right)\left(x^{\{m\}}\otimes I_{r}\right)}$ was introduced in order to establish whether a matrix form is SOS via an LMI.⁹

Consider the free algebra R⟨X⟩ generated by the n noncommuting letters X = (X₁, ..., X_n) and equipped with the involution ^T, such that ^T fixes R and X₁, ..., X_n and reverses words formed by X₁, ..., X_n. We consider Hermitian noncommutative polynomials f, which are the noncommutative polynomials of the form f = f^T. When evaluating a Hermitian noncommutative polynomial f on any n-tuple of real matrices of any size r × r produces in a positive semi-definite matrix, f is said to be matrix-positive.

A noncommutative polynomial is SOS if there exists noncommutative polynomials ${\displaystyle h_{1},\ldots ,h_{k}}$ such that $${\displaystyle f(X)=\sum _{i=1}^{k}h_{i}(X)^{T}h_{i}(X).}$$

Surprisingly, in the noncommutative scenario a noncommutative polynomial is SOS if and only if it is matrix-positive.¹⁰ Moreover, there exist algorithms available to decompose matrix-positive polynomials in sum of squares of noncommutative polynomials.¹¹

A complex polynomial ${\displaystyle p(z_{1},\dots ,z_{n})}$ in variables ${\displaystyle z_{1},\dots ,z_{n}}$ and their conjugates ${\displaystyle z_{1}^{*},\dots ,z_{n}^{*}}$ is Hermitian if it takes on only real values, or equivalently if each term has an equal number of conjugated and un-conjugated variables. It is a hermitian sum-of-squares (HSOS) if there are complex polynomials ${\displaystyle g_{1},\dots ,g_{k}}$, in only the un-conjugated variables ${\displaystyle z_{1},\dots ,z_{n}}$, such that $${\displaystyle p(z)=\sum _{i=1}^{k}g_{i}^{*}(z)g_{i}(z).}$$ A Hermitian square matricial representation of ${\displaystyle p}$ is a matrix ${\displaystyle M}$ such that ${\displaystyle p(z)=(z^{\{m\}})^{*}Mz^{\{m\}}}$, where ${\displaystyle (z^{\{m\}})^{*}}$ is the Hermitian transpose of the vector ${\displaystyle z^{\{m\}}}$. As first observed by Putinar, there is a choice of the matrix ${\displaystyle M}$ having certain symmetry properties, which is in fact unique and can be written explicitly. Thus testing if a Hermitian polynomial is HSOS of degree ${\displaystyle m}$ can be done by testing if a single, fixed matrix is positive-semidefinite.¹²