Support setting starting values

[PaioPaio]

Hello :)

I've seen this old thread on discourse https://discourse.julialang.org/t/warm-start-mosek-in-a-linear-polynomial-optimization-problem/90478 and since there wasn't an open issue I obliged.
I'm planning to translate a bit of MATLAB code which I've written using SOSTOOLS to Julia using SumOfSquares.jl. In particular the code uses an iterative algorithm to approximate the RAS of a non-linear system.
It would be nice to be able to warm start the SOS optimization at each iteration using the polynomial obtained from the previous step.

[blegat]

Sure, let's do it then :) Note that Mosek does not support starting values. SCS/COSMO do support it though

[votroto]

Is this still broken? For the QCQP this can be quite important too -- along with propagating the start values of monomials to the lifted variables.

[blegat]

It's not the same as QCQP, it should work for QCQP after #115

[votroto]

Are you sure? Compare Gurobi's output on the projection done by hand

using JuMP, PolyJuMP, Gurobi

m = Model(Gurobi.Optimizer)
@variable(m, x >= 0, start=1/sqrt(3))
@variable(m, y >= 0, start=1-start_value(x))
@variable(m, xx, start=start_value(x)*start_value(x))

@constraint(m, xx == x*x)
@constraint(m, x + y == 1)

@objective(m, Min, xx*x + y)
optimize!(m)

versus the current implementation

m = Model(() -> PolyJuMP.QCQP.Optimizer(Gurobi.Optimizer()))
@variable(m, x >= 0, start=1/sqrt(3))
@variable(m, y >= 0, start=1-start_value(x))

@constraint(m, x + y == 1)

@objective(m, Min, x*x*x + y)
optimize!(m)

The first one explicitly writes Loaded user MIP start with objective 0.6151 , while the second one does not.

I believe it would be necessary to track the variables in the QCQP.Optimizer though, as the variables get copied to the model after final_touch. That might not be the worst thing, maybe the polyvar/moivar maps should be kept in the optimizer too, instead of the ScalarPolynomialFunction objects.

[blegat]

@votroto #119 should fix this