About the default objective function used in non-parametrized SoS problems

[PoeticExposition]

The default objective function appears to be $0$ when no parameters (i.e., “JuMP's decision variables which are created using the @variable macro”) are present. However, it is said that “using the trace as the objective function is a heuristic for obtaining sum-of-squares decompositions with a small number of squares”. Does there exist a way to use the trace of the corresponding Gram matrix as the objective function automatically?

[blegat]

Good question. When you use @variable with SOSPoly, they get a GramMatrix back. You can get use this to get the symmetric matrix:

SumOfSquares.jl/src/gram_matrix.jl

Line 112 in 2c40b50

MultivariateMoments.value_matrix(p::GramMatrix{T}) where {T} = p.Q

And then you can use the trace of that matrix.

[PoeticExposition]

Thank you for your advice. But what if there is no need to use SOSPoly?
The code in a trivial SOS decomposition example (although this example is overly trivial, I believe that it should illustrate the point) is as follows:

using DynamicPolynomials, SumOfSquares
import CSDP
@polyvar x y;
p = x^2 - x*y^2 + y^4 + 1;
model = SOSModel(CSDP.Optimizer);
@constraint(model, cref, p in SOSCone());
#optimize!(model);

In this case, SOSPoly is not used at all; can we still use the trace of the corresponding Gramian as the objective function before invoking optimize!?