Problems with Projection onto the SOS Cone

[JOlucak]

We are interested to solve the projection onto the SOS (or any other cone) to obtain the constraint violation. In CaSoS this can be easily done for some constraint e.g.

s    = casos.PS.sym('q',grambasis( constraint ) );

   % projection error
    e = s - constraint;
    
    
    % define SOS problem 
    %   min ||s-p||^2  s.t. s is SOS
    proj_sos = struct('x',s,...
                               'f',dot(e,e);
    
    opts.Kx            = struct('sos', length(s));
    opts.error_on_fail = 1;
    obj.projConPara    =  casos.sossol('S','mosek',proj_sos,opts);

However, if Mosek is used as solver, it frequently runs into numerical issues.
One could "ignore" this by simply setting opts.error_on_fail = 0 what is obviously not recommended.
Also re-writting the problem with linear decision variables and additional SOS constraint might not help here.

This issue needs furhter investigation.

Points to be clarified:

• The computation of the projection should be always possible?
• How trustworthy is the mosek "result" in case of numerical problems? Obviously, mosek has problem in solving the problem but the question is where these problems come from.
• Which alternatives do we have?

[tcunis]

The computation of the projection should be always possible?

Yes, the projection onto a convex set (cone) is a strictly convex problem and hence always has a (unique) solution.

How trustworthy is the mosek "result" in case of numerical problems?

You should check first if the solution Mosek returns for x is feasible (i.e., sum-of-squares). If it is not, the result is not to be trusted. If it is, you could check whether constraint is sum-of-squares.

Which alternatives do we have?

Possible alternatives:

1. Other solver: SCS supports quadratic objective functions natively, thus avoiding the epigraph formulation.
2. Other objective: Project using L1 or Linf norm, again avoiding the quadratic cost function and its epigraph formulation.
3. Scaling: It is possible the problem is illconditioned, possibly due to scaling of the polynomial coefficients. You could try using pnorm2 for the objective function which defines an L2 norm on the polynomials via an integral.