Equality constraints in SOS

[Matanya Horowitz]

I have an application where I would like to enforce equality between several optimization polynomials over a semialgebraic set. By application of the Positivstellensatz, this can be enforced as follows.

V1(x) = V2(x) on the set p(x)=0 if there exists polynomial c(x) such that

V1(x)-V2(x) - c(x)p(x) = 0

An example of this theorem's use is available in "Analysis of Switched and hybrid systems - beyond piecewise quadratic methods" by Prajna and Papachristodoulou, Theorem 3.

My question is, is there a way to enforce equality constraints in SOS programs in Yalmip? In the examples and documentation, >= constraints are relaxed as SOS constraints. It's clear that I could introduce some equality error epsilon and then make sure that the error between |V1 - V2| <= eps, but I was wondering if there was a more direct way. I could also get out the SDP variables and solve it as an SDP with my own constraints, but I'd like to take advantage of Yalmip's pre- and post-processing.

An alternative way of saying this is, is there an analogue to the soseq command in SOSTools?

My thanks to those who respond.

[Johan Löfberg]

If you want the polynomial V1(x)-V2(x) - c(x)p(x)  to be the zero polynomial, and c(x) is parameterized by some coefficients coeff, you would add the constraint coefficients(V1-V2-c*p,coeff)==0

[Matanya Horowitz]

Ah, of course. Exactly what I was looking for, thank you very much.