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| {{about|representing polynomial as sum of squares of polynomials|representing polynomial as a sum of squares of rational functions|Hilbert's seventeenth problem|the sum of squares of consecutive integers|square pyramidal number|representing an integer as a sum of squares of integers|Lagrange's four-square theorem}}
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| In [[mathematics]], a [[Homogeneous polynomial|form]] (i.e. a homogeneous polynomial) ''h''(''x'') of degree 2''m'' in the real ''n''-dimensional vector ''x'' is sum of squares of forms (SOS) if and only if there exist forms <math>g_1(x),\ldots,g_k(x)</math> of degree ''m'' such that
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| :<math>
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| h(x)=\sum_{i=1}^k g_i(x)^2 .
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| </math>
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| Explicit sufficient conditions for a form to be SOS have been found.<ref>[http://www.optimization-online.org/DB_HTML/2007/02/1587.html], [http://www.mathcs.emory.edu/~vicki/pub/sos.pdf].</ref> However every real nonnegative form can be approximated as closely as desired (in the <math>l_1</math>-norm of its coefficient vector) by a sequence of forms <math>\{f_\epsilon\}</math> that are SOS.<ref>[http://portal.acm.org/citation.cfm?id=1330215.1330223&coll=GUIDE&dl=].</ref>
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| == Square matricial representation (SMR) ==
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| To establish whether a form ''h''(''x'') is SOS amounts to solving a [[convex optimization]] problem. Indeed, any ''h''(''x'') can be written as
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| :<math>
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| h(x)=x^{\{m\}'}\left(H+L(\alpha)\right)x^{\{m\}}
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| </math> | |
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| where <math>x^{\{m\}}</math> 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
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| :<math> | |
| h(x)=x^{\left\{m\right\}'}Hx^{\{m\}}
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| </math>
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| and <math>L(\alpha)</math> is a linear parameterization of the [[vector space|linear space]]
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| :<math>
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| \mathcal{L}=\left\{L=L':~x^{\{m\}'} L x^{\{m\}}=0\right\}.
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| </math>
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| The dimension of the vector <math>x^{\{m\}}</math> is given by
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| :<math>
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| \sigma(n,m)=\binom{n+m-1}{m}
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| </math>
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| whereas the dimension of the vector <math>\alpha</math> is given by
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| :<math>
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| \omega(n,2m)=\frac{1}{2}\sigma(n,m)\left(1+\sigma(n,m)\right)-\sigma(n,2m).
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| </math>
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| Then, ''h''(''x'') is SOS if and only if there exists a vector <math>\alpha</math> such that
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| :<math>
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| H + L(\alpha) \ge 0,
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| </math> | |
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| meaning that the matrix <math>H + L(\alpha)</math> is [[positive-semidefinite matrix|positive-semidefinite]]. This is a [[linear matrix inequality]] (LMI) feasibility test, which is a convex optimization problem. The expression <math>h(x)=x^{\{m\}'}\left(H+L(\alpha)\right)x^{\{m\}}</math> 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).
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| == Examples ==
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| <ul>
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| <li>Consider the form of degree 4 in two variables <math>h(x)=x_1^4-x_1^2x_2^2+x_2^4</math>. We have
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| :<math> | |
| m=2,~x^{\{m\}}=\left(\begin{array}{c}x_1^2\\x_1x_2\\x_2^2\end{array}\right),
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| ~H+L(\alpha)=\left(\begin{array}{ccc}
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| 1&0&-\alpha_1\\0&-1+2\alpha_1&0\\-\alpha_1&0&1
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| \end{array}\right).
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| </math>
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| Since there exists α such that <math>H+L(\alpha)\ge 0</math>, namely <math>\alpha=1</math>, it follows that ''h''(''x'') is SOS.
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| <li>Consider the form of degree 4 in three variables <math>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</math>. We have
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| :<math>
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| 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),
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| ~H+L(\alpha)=\left(\begin{array}{cccccc}
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| 2&-1.25&0&-\alpha_1&-\alpha_2&-\alpha_3\\
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| -1.25&2\alpha_1&0.5+\alpha_2&0&-\alpha_4&-\alpha_5\\
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| 0&0.5+\alpha_2&2\alpha_3&\alpha_4&\alpha_5&-1\\
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| -\alpha_1&0&\alpha_4&5&0&-\alpha_6\\
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| -\alpha_2&-\alpha_4&\alpha_5&0&2\alpha_6&0\\ | |
| -\alpha_3&-\alpha_5&-1&-\alpha_6&0&1
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| \end{array}\right).
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| </math> | |
| Since <math>H+L(\alpha)\ge 0</math> for <math>\alpha=(1.18,-0.43,0.73,1.13,-0.37,0.57)</math>, it follows that ''h''(''x'') is SOS.
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| </ul> | |
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| == Matrix SOS ==
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| 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 <math>G_1(x),\ldots,G_k(x)</math> of degree ''m'' such that
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| :<math>
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| F(x)=\sum_{i=1}^k G_i(x)'G_i(x) .
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| </math>
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| == Matrix SMR ==
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| 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
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| :<math>
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| F(x)=\left(x^{\{m\}}\otimes I_r\right)'\left(H+L(\alpha)\right)\left(x^{\{m\}}\otimes I_r\right)
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| </math>
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| where <math>\otimes</math> is the [[Kronecker product]] of matrices, ''H'' is any symmetric matrix satisfying
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| :<math>
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| F(x)=\left(x^{\{m\}}\otimes I_r\right)'H\left(x^{\{m\}}\otimes I_r\right)
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| </math>
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| and <math>L(\alpha)</math> is a linear parameterization of the linear space | |
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| :<math>
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| \mathcal{L}=\left\{L=L':~\left(x^{\{m\}}\otimes I_r\right)'L\left(x^{\{m\}}\otimes I_r\right)=0\right\}.
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| </math>
| |
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| The dimension of the vector <math>\alpha</math> is given by
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| :<math>
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| \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).
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| </math>
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| | |
| Then, ''F''(''x'') is SOS if and only if there exists a vector <math>\alpha</math> such that the following LMI holds:
| |
| | |
| :<math>
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| H+L(\alpha) \ge 0.
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| </math> | |
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| The expression <math>F(x)=\left(x^{\{m\}}\otimes I_r\right)'\left(H+L(\alpha)\right)\left(x^{\{m\}}\otimes I_r\right)</math> was introduced in [3] in order to establish whether a matrix form is SOS via an LMI.
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| == References ==
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| [1] G. Chesi, A. Tesi, A. Vicino, and R. Genesio, ''On convexification of some minimum distance problems'', 5th European Control Conference, Karlsruhe (Germany), 1999.
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| [2] M. Choi, T. Lam, and B. Reznick, ''Sums of squares of real polynomials'', in Proc. of Symposia in Pure Mathematics, 1995.
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| [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.
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| | |
| ==Notes==
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| {{Reflist}}
| |
| | |
| [[Category:Homogeneous polynomials]]
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| [[Category:Real algebraic geometry]]
| |
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