Moment Tools

The package provide tools for moment optimization problems on Positive Moment Sequences (PMS).

A PMS is a sequence of moments $\mu=(\mu_{\alpha})$ or equivalently a linear functional $\mu: p \in \mathbb{R}[\mathbf{x}] \mapsto \langle \mu, p \rangle = \sum_{\alpha} p_{\alpha} \mu_{\alpha}$, which is positive on the square of the polynomials: $\langle \mu, p^2 \rangle \geq 0$ for all $p\in \mathbb{R}[\mathbf{x}]$.

Optimization

Optimization problems of the following form are considered:

\[\begin{array}{rl} \mathrm{inf}_{\mu_i \in PMS} & \sum_i \langle f_i\star \mu_i, 1 \rangle \\ s.t. & \sum_i g_{i,j}\star \mu_i \succeq 0, \quad j=1,\ldots, n_1 \\ & \sum_i h_{i,j}\star \mu_i = 0, \quad j=1,\ldots, n_2\\ & \sum_i \langle p_{i,j}\star \mu_i, 1 \rangle \ge 0 , \quad j=1,\ldots, n_3\\ & \sum_i \langle q_{i,j}\star \mu_i, 1 \rangle = 0, \quad j=1,\ldots, n_4 \\ \end{array}\]

where

$\mu_i$ are Positive Moment Sequences,
$f_i, g_{i,j}, h_{i,j}, p_{i,j}, q_{i,j} \in \mathbb{R}[\mathbf{x}]$ are multivariate polynomials.

The solution of such optimization problem is approximated by the solution of a truncated relaxation of the problem, which is a convex optimization problem on Positive SemiDefinite matrices. Tools to construct such moment relaxation of a given order are available in the package.

Decomposition

Decomposition tools are available to decompose or approximate a PMS by a weighted sum of Dirac measures:

\[\mu \approx \sum_k \omega_k \, \delta_{\xi_k}\]

where $\omega_k\in \mathbb{R}$ (resp. $\mathbb{C}$), $\xi_k \in \mathbb{R}^n$ (resp. $\mathbb{C}^n$) and $\delta_{\xi}$ is the Dirac measure at the point $\xi$.