Obtaining Symmetric Positive Semi-Definite Matrices from Sum of Squares Variables

Question

I am a bachelor student and have the following question.

I am currently working on converting constraint functions into new constraints as sum of squares since I would like to solve the optimization problem through SDP-solvers.

Thanks to SOS-Tools in MATLAB I have obtained the following sum of squares variables:

$s=(238.3)t^4-(425.3)t^3+(223.7)t^2-(20.63)t+1.0$

$q=(50.84)t^2-(18.86)t+9.438$

where $t$ is an independent variable.

What I need is the $S$ & $Q$ Matrices where

$[1, t, t^2]\cdot S\cdot[1, t, t^2]T = s$

$[1, t]\cdot Q \cdot[1, t]T = q$

$S,Q \ge 0$

The next logical step would be to use another function of SOS-Tools and find such matrices with findsos(s) or findsos(q). This function; however, returns for some reason wrong matrices.

If the equations are calculated by hand, one can obtain the $Q$ matrix.

$q = Q(2,2) \cdot t^2 + 2 \cdot Q(1,2) \cdot t + Q(1,1)$

It's a little bit tricky for the other one.

$s = S(3,3) \cdot t^4 + 2\cdot S(2,3) \cdot t^3 + 2 \cdot (S(1,3) + S(2,2)) \cdot t^2 + 2\cdot S(1,2) \cdot t + S(1,1)$

$S(1,3)$ and $S(2,2)$ have to be decided in such a way that the matrix $(S)$ will be positive semi-definite.

As my problem gets more complex, the size of the matrices will also get bigger. That's why it is important for me to generalize things. Does anybody happen to know a trick on how to do it?

I'd very much appreciate your answer.

Kind regards

Answer 1

You have not explained in what sense they are wrong. I strongly doubt they are as this really is a trivial SOS problem.

As an alternative solution to double-check your results, here is the solution computed using YALMIP instead (disclaimer, MATLAB toolbox written by me).

sdpvar t
s = 238.3*t^4-425.3*t^3+223.7*t^2-20.63*t+1.0;
q = 50.84*t^2-18.86*t+9.438;
[~,~,Grams] = solvesos([sos(s), sos(q)])
S = Grams{1};
Q = Grams{2};
S =

    1.0000  -10.3150    8.2833
  -10.3150  207.1334 -212.6500
    8.2833 -212.6500  238.3000

Q =
    9.4380   -9.4300
   -9.4300   50.8400