A simple univariate example

using DynamicPolynomials, MomentPolynomialOpt

X = @polyvar x
p = x^4+x^3-x-1

$ -1 - x + x^{3} + x^{4} $

We minimize $x^2$ under the constraint $p=0$.

using JuMP, MosekTools; mpo_optimizer(Mosek.Optimizer, "QUIET" => true)
v, M = minimize(x^2, [p], [], X, 4)

(0.9999999937113259, A JuMP Model
Minimization problem with:
Variables: 9
Objective function type: VariableRef
`AffExpr`-in-`MathOptInterface.EqualTo{Float64}`: 6 constraints
`Vector{AffExpr}`-in-`MathOptInterface.PositiveSemidefiniteConeTriangle`: 1 constraint
Model mode: AUTOMATIC
CachingOptimizer state: ATTACHED_OPTIMIZER
Solver name: Dual model with Mosek attached
Names registered in the model: mu, type)

We use Mosek optimizer to solve the underlying SDP optimization problem. The first argument $x^2$ of the function minimize is the objective function. The second argument is the array of equality constraints, the third is the array of non-negativity constraints (here empty).

The function minimize returns v the optimal value found (here it is $\approx 1$) and M the Moment Model built for the optimization (which type is MomentPolynomialOpt.MOM.Model).

The minimizers can be obtained from M as follows:

get_minimizers(M)

1×2 Matrix{Float64}:
 -1.0  1.0

The minimizers are presented as a $n\times r$ matrix, where $n$ is the number of variables (here $n=1$) and $r$ is the number of points (here $r=2$).

An approximation of the associate measure can be obtained as follows:

get_measure(M)

([0.4781556356477239, 0.5218443598681404], [0.9999999995837139 -1.0000000010811763])

It corresponds approximately to the measure $0.5\, \delta_{-1} + 0.5\, \delta_{1}$ where $\delta_x$ is the Dirac measure supported on $x$.

Next, we search the root(s) with $x \geq 0$. For that purpose, we minimize nothing (i.e. $1$) and add the non-negativity constraint $x\geq 0$:

v, M = minimize(1, [p], [x], X, 4)
get_minimizers(M)

1×1 Matrix{Float64}:
 1.0000000000608462

Now we search the negative root with maximal value of $x$.

v, M = maximize(x, [p], [-x], X, 4)
get_minimizers(M)

1×1 Matrix{Float64}:
 -0.9999999993829266