Moment Program

moments(M, X, d::Int, symb::Symbol...)

Define variables for the moments of the monomials of degree less than d in the variables X. If

• symb == :PSD, non-negativity constraints are added;
• symb == :PRB, unit mass and non-negativity constraints are added.

M = MOM.Model( `sense`, f, H, G, X, d)

Construct the Moment Program in the variables X of order d.

• sense == :inf or :sup
• f polynomial objective function
• H =[h1, h2, ...] array of polynomial equality constraints (can be empty)
• G =[g1, g2, ...] array of non-negativity constraints (can be empty)
• X is the vector of variables
• d is the order of the moment relaxation.

M = MOM.Model(C, X, d)

Construct the Moment Program where

• C is a vector of pairs (f, sense ) of objective or constraints where f is a polynomial and sense is "inf", "min", "sup", "max", ">=0", "<=0", "=0", or an interval
• X is the vector of variables
• d is the order of the moment relaxation.

add_constraint_zero(M, eq, mu::Moments)

Add to the moment program M, the constraints $eq\star \mu = 0$, that is all the constraints $\mu(p*m)==0$ for $degree(m) \le 2*degree(M)-degree(p)$.

add_constraint_nneg(M, mu::Moments, g)

Add to the moment program M, the constraints $g \star \mu \succeq 0$, that is the moment matrix of $g\star \mu$ is PSD.

add_constraint_moment(M, mu::Moments,  p => v)

Add to the moment program M, the constraint $\mu(p) - v == 0$.

add_constraint_unitmass(M, mu)

Add to the moment program M, the constraint $\langle \mu, 1 \rangle - 1 = 0$.

set_objective(M, sense, p, mu)

Set the "inf" or "sup" objective function to $\sum_{i=1}^{\nu} \langle p\star \mu_i, 1 \rangle$ where p is a polynomial.

set_objective_tv(M)

Set the objective function of the moment program to the total variation of the moment sequence $\mu$, that is the sum of the unit mass of the positive moment sequences $\sum_{i=1}^{\nu} \langle \mu_i, 1\rangle$.

set_objective_ncl(M)

Set the objective function of moment program to the nuclear norm or equivalently the trace of the moment matrices.