SOS optimization/ reformulate symbolic expresions

1

Hey there,

I want to check if a given polynomial p(x) is a Sum-of-Squares polynomial; therefore, I need to find a positive semi-definite matrix Q such that p(x) = z(x)' Q z(x) where z(x) is a vector containing monomials in x.

So, I would like to create a symbolic vector z(x) = [1 x x^2 ...]' and invoke the optimization problem like the following:

syms x
z =  z(x) = [1;  x; x^2]

cvx_begin

    n = size(z)
    variables Q(n,n) SDP

    subject to

    p(x)  == z(x)' Q z(x)

cvx_end

I do know that the equality does not work as syms and cvx objects are not compatible. However, I wonder if someone has a trick to circumvent this issue, or even better, if someone already implemented SOS optimization on cvx.

Thank you for your help and greetings,

Maurits

ps.: yeah yeah, there is SOSTOOLS but I need DSOS; and the lingo in spotlessSOS is rather hard.

2

I don’t know if you’ll find anything on SOS for CVX.

However, YALMIP has an SOS framework https://yalmip.github.io/tutorial/sumofsquaresprogramming/ .

3

I didn’t and thus settled for the spotless package. Though the overall package is rather undocumented, in the files forked by majumdar is an implementation example for DSOS.