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

Every form that is SOS is also a positive polynomial, and although the converse is not always true, Hilbert proved that for n = 2, 2m = 2, or n = 3 and 2m = 4 a form is SOS if and only if it is positive.^[1] The same is also valid for the analog problem on positive symmetric forms.^[2][3]

Although not every form can be represented as SOS, explicit sufficient conditions for a form to be SOS have been found.^[4][5] 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]

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

where ${\displaystyle x^{\{m\}}}$ 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 and ${\displaystyle L(\alpha )}$ is a linear parameterization of the linear space

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

whereas the dimension of the vector ${\displaystyle \alpha }$ is given by

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

meaning that the matrix ${\displaystyle H+L(\alpha )}$ is positive-semidefinite. This is a linear matrix inequality (LMI) feasibility test, which is a convex optimization problem. The expression ${\displaystyle h(x)=x^{\{m\}'}\left(H+L(\alpha )\right)x^{\{m\}}}$ was introduced in ^[7] 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.^[8]

Examples

• Consider the form of degree 4 in two variables ${\displaystyle h(x)=x_{1}^{4}-x_{1}^{2}x_{2}^{2}+x_{2}^{4}}$. We have
  $${\displaystyle m=2,~x^{\{m\}}={\begin{pmatrix}x_{1}^{2}\\x_{1}x_{2}\\x_{2}^{2}\end{pmatrix}}\!,~H+L(\alpha )={\begin{pmatrix}1&0&-\alpha _{1}\\0&-1+2\alpha _{1}&0\\-\alpha _{1}&0&1\end{pmatrix}}\!.}$$

  
  Since there exists α such that ${\displaystyle H+L(\alpha )\geq 0}$, namely ${\displaystyle \alpha =1}$, it follows that h(x) is SOS.
• Consider the form of degree 4 in three variables ${\displaystyle h(x)=2x_{1}^{4}-2.5x_{1}^{3}x_{2}+x_{1}^{2}x_{2}x_{3}-2x_{1}x_{3}^{3}+5x_{2}^{4}+x_{3}^{4}}$. We have
  $${\displaystyle m=2,~x^{\{m\}}={\begin{pmatrix}x_{1}^{2}\\x_{1}x_{2}\\x_{1}x_{3}\\x_{2}^{2}\\x_{2}x_{3}\\x_{3}^{2}\end{pmatrix}},~H+L(\alpha )={\begin{pmatrix}2&-1.25&0&-\alpha _{1}&-\alpha _{2}&-\alpha _{3}\\-1.25&2\alpha _{1}&0.5+\alpha _{2}&0&-\alpha _{4}&-\alpha _{5}\\0&0.5+\alpha _{2}&2\alpha _{3}&\alpha _{4}&\alpha _{5}&-1\\-\alpha _{1}&0&\alpha _{4}&5&0&-\alpha _{6}\\-\alpha _{2}&-\alpha _{4}&\alpha _{5}&0&2\alpha _{6}&0\\-\alpha _{3}&-\alpha _{5}&-1&-\alpha _{6}&0&1\end{pmatrix}}.}$$

  
  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.

Generalizations

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 ${\displaystyle G_{1}(x),\ldots ,G_{k}(x)}$ of degree m such that

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

where ${\displaystyle \otimes }$ is the Kronecker product of matrices, H is any symmetric matrix satisfying and ${\displaystyle L(\alpha )}$ is a linear parameterization of the linear space

The dimension of the vector ${\displaystyle \alpha }$ is given by

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

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

Noncommutative polynomial SOS

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. By analogy with the commutative case, the noncommutative symmetric polynomials f are the noncommutative polynomials of the form f = f^T. When any real matrix of any dimension r × r is evaluated at a symmetric noncommutative polynomial f results 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

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]

References

See also

• Sum-of-squares optimization
• Positive polynomial
• Hilbert's seventeenth problem
• SOS-convexity