Working with monomial vectors

[Fabian-Geyer]

Before introducing the sparsity concept for the monomial vectors, working with them could sometimes be more intuitive (from a beginners perspective). Consider the literature example from [1] where one could write:

x = casos.PS.sym('x', 2);

% create monomials vectors
z_1 = monomials(x, [1:1]);
z_2 = monomials(x, 2:2);
z_3 = monomials(x, 3:3);

% dynamics
f = A_1*z_1 + A_2*z_2 + A_3*z_3;

It might not be "best practice", but is there another intuitive way of obtaining something similar with the newer version?

@tcunis / @renatoloureiro
[1] P. Seiler and G. J. Balas, “Quasiconvex sum-of-squares programming,” in 49th IEEE Conference on Decision and Control (CDC), Atlanta, GA, USA: IEEE, Dec. 2010, pp. 3337–3342. doi: 10.1109/CDC.2010.5717672.

[JOlucak]

One possible way is:

% generate random polynomial
x = casos.PS('x',2);
mon = monomials(x,0:2);
a = rand(mon.nnz,1)
poly = casos.PS(,a)

In the above case

% create monomials vectors
z_1 = monomials(x, [1:1]);
z_2 = monomials(x, 2:2);
z_3 = monomials(x, 3:3);

% dynamics
A1 = rand(z_1.nnz,1)
A2 = rand(z_2.nnz,1)
A3 = rand(z_3.nnz,1)

% f = A_1z_1 + A_2z_2 + A_3*z_3;
f = casos.PS(z_1,A1) + casos.PS(z_2,A2) + casos.PS(z_3,A3)

[tcunis]

this works provided that A1 through A3 are vectors which, in the example of Seiler and Balas, they're not.

[tcunis]

It is important to note that the monomial sparsity pattern is fundamentally different from a vector of monomials. The latter is a vector-valued polynomial which has an onto-relation between monomials and entries (often in a sorted manner, i.e., $z_\alpha = x^\alpha$). The monomial sparsity pattern describe the monomials in a compact form and is, here, scalar-valued.

In sosopt, the monomials method is used to create a vector of monomials as sosopt does not have a sparsity concept. I have kept the syntax but return a monomial sparsity pattern; this is motivated from the following use case:

z = monomials(x, degs);
s = casos.PS.sym('q',z);

In fact, the monomials method is internally mapped to casos.Sparsity.scalar.

[tcunis]

currently, the only way to implement the example is to use the to_vector method for sparsity patterns, then rely on the implicit casting from casos.Sparsity to casos.PD. This effectively converts a monomial sparsity pattern into a vector of monomials:

z1 = to_vector(monomials(x,1));
...
z3 = to_vector(monomials(x,3));

However, observe that the variables z1 through z3 are still of type casos.Sparsity and the implicit casting might not be desirable.

[tcunis]

Alternatively, I suggest to overload the mpower operator for indeterminate variables to return a vector of monomials (of type casos.PD). Then, the example could look like

f = A1*x + A2*x^2 + A3*x^3;