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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
of degree m such that
![{\displaystyle h(x)=\sum _{i=1}^{k}g_{i}(x)^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/393576199445ebae5cc21be5aed8984c8e3faf35)
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
-norm of its coefficient vector) by a sequence of forms
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
![{\displaystyle h(x)=x^{\{m\}'}\left(H+L(\alpha )\right)x^{\{m\}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cbda35064e92cdbd6cfc023e9869f231142695e)
where
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
![{\displaystyle h(x)=x^{\left\{m\right\}'}Hx^{\{m\}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/535266babcc6edf297c2a47c80e0421cd1626187)
and
is a linear parameterization of the linear space
![{\displaystyle {\mathcal {L}}=\left\{L=L':~x^{\{m\}'}Lx^{\{m\}}=0\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a01e91e36eaabaf4357c96a31dfe20d72906bec5)
The dimension of the vector
is given by
![{\displaystyle \sigma (n,m)={\binom {n+m-1}{m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c70c37b819dd5167914b5fab5933c5a6b418b83)
whereas the dimension of the vector
is given by
![{\displaystyle \omega (n,2m)={\frac {1}{2}}\sigma (n,m)\left(1+\sigma (n,m)\right)-\sigma (n,2m).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48bca70d27dfd44bdf0906b8bbc33110b8a903dd)
Then, h(x) is SOS if and only if there exists a vector
such that
![{\displaystyle H+L(\alpha )\geq 0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a38a83404919d70a72b9d4f84ac7a6f11481957)
meaning that the matrix
is positive-semidefinite. This is a linear matrix inequality (LMI) feasibility test, which is a convex optimization problem. The expression
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
. We have
![{\displaystyle m=2,~x^{\{m\}}=\left({\begin{array}{c}x_{1}^{2}\\x_{1}x_{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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7817a32fc133e861bacbef204e8cb07ad06ac8f8)
Since there exists α such that
, namely
, it follows that h(x) is SOS.
- Consider the form of degree 4 in three variables
. We have
![{\displaystyle m=2,~x^{\{m\}}=\left({\begin{array}{c}x_{1}^{2}\\x_{1}x_{2}\\x_{1}x_{3}\\x_{2}^{2}\\x_{2}x_{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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4552ba877946c01b3284fa8d75c2508e4ddbd83)
Since
for
, 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
of degree m such that
![{\displaystyle F(x)=\sum _{i=1}^{k}G_{i}(x)'G_{i}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10b91e2a687a210327c554f5cb5129de776ae91f)
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
![{\displaystyle F(x)=\left(x^{\{m\}}\otimes I_{r}\right)'\left(H+L(\alpha )\right)\left(x^{\{m\}}\otimes I_{r}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8798540bca6bbf1ba1d285e76178622a3c5705)
where
is the Kronecker product of matrices, H is any symmetric matrix satisfying
![{\displaystyle F(x)=\left(x^{\{m\}}\otimes I_{r}\right)'H\left(x^{\{m\}}\otimes I_{r}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4578d749d87bea10bddb65cc48222438502b896e)
and
is a linear parameterization of the linear space
![{\displaystyle {\mathcal {L}}=\left\{L=L':~\left(x^{\{m\}}\otimes I_{r}\right)'L\left(x^{\{m\}}\otimes I_{r}\right)=0\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abc96c78eba6f9e86d68743ce34d96221b8d40b8)
The dimension of the vector
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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/192b69d876a819d346809cfbec3558b47ae81d2c)
Then, F(x) is SOS if and only if there exists a vector
such that the following LMI holds:
![{\displaystyle H+L(\alpha )\geq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/252463bebd22832c4003fcd0d97855b8e0bfb12a)
The expression
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
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.