Optimization

v, M = JuMP.set_optimizer(M, optimizer)

Set the optimizer of the moment program M to the dual optimizer of optimizer.

v, M = minimize(f, [e1, e2, ...], [p1, p2, ...], X, d, optimizer)

Compute the infimum of f under the constraints $e_i$ =0 and $p_i \geq 0$ using a relaxation of order d on the moments in the variable X and the optimizer M.

$f, e_i, p_i$ should be polynomials in the variables X.

If the problem is feasible and has minimizers, it outputs

• v: the minimum value
• M: the moment model of type MOM.Model

Example

using MomentTools

X  = @polyvar x1 x2
e1 = x1^2-2
e2 = (x2^2-3)*(x1*x2-2)
p1 = x1
p2 = 2-x2
v, M = minimize(-x1, [e1, e2], [p1, p2], X, 3)

To recover the optimal values, see get_minimizers, get_measure, get_series.

v, M = polar_minimize(f, [h1, h2, ...], [g1, g2, ...], X, degree_approx, optimizer)

Compute the infimum of the f (equality constraints hi == 0 and the sign constraints gi >= 0) on the moment side. It does that calling minimize(...), replacing the equality constraints [h1, h2, ...] with generators of the polar ideal.

f, hi, gi should be polynomials in the variables X.

If the problem is feasible and has minimizers, it outputs

• v: the minimum value
• M: the moment model of type MOM.Model

v, M = maximize(f, [e1, e2, ...], [p1, p2, ...], X, d, optimizer)

Similar to the function minimize but compute the supremun of f.

v, M = optimize(M)

Run the optimizer on the moment program M and output the objective_value v and the moment program M. If the optimization program has no value, it returns nothing and M.

v, M = optimize([(f, set), ...], X, d, optimizer)

Solve the moment program of relaxation of order d in the variables X, defined by the constraint or objective paires (f, set) where f is a polynomial and set is a string

• "inf", "sup" to define the objective.
• "=0" to define zero constraints:
• ">=0", "<=0" to define sign constraints

It outputs

• v: the optimal value
• M: the optimized moment program of type MOM.Model

Example

using MomentTools

X  = @polyvar x1 x2
e1 = x1^2-2
e2 = (x2^2-3)*(x1*x2-2)
p1 = x1
p2 = 2-x2
v, M = optimize([(-x1, "inf"), (e1, "=0"), (e2, "=0"), (p1, ">=0"), (p2>=0)], X, 3)

To recover the optimal values, see get_minimizers, get_measure, get_series.

get_minimizers(M)

Return the minimizer points of the optimized moment program M.

get_minimizer(M)

[1.41421 1.73205; 1.41421 1.41421; 1.41421 -1.73205]

w, Xi = get_measure(M, lambda = [(-1)^(k-1) for k in 1:M[:nu]])

Return the approximation of the moment sequence $\sum_{i=1}^{\nu} \lambda_i \mu_i$ as weighted sum of Dirac measures: $\sum_{k=1}^{r} \omega_k \delta_{\xi_k}$ where

• w is the vector of weights of the Dirac measures.
• Xi is matrix of $n\times r$ support points of the corresponding Dirac measures. The column Xi[:,i] is the support point $\xi_{i}$ of the ith Dirac measure and its weights is w[i].

w, Xi = get_measure(M)

([0.1541368146508854, 0.5889741915171074, 0.256888993597116], [1.4142135624216647 1.414213562080608 1.4142135620270329; -1.732052464639053 1.4141771454788292 1.7319839273833693])

get_series(M)

Return the vector of $\nu$=M[:nu] series of optimal moments of the optimized moment program M.