The authoritative documentation is the PDF document gposolver-VERSION.pdf, located in the directory doc in the distribution archive of the respective GpoSolver version. The documentation for the latest version of GpoSolver can be downloaded from here:

Let $p(\mathbf{x})$ , $g_{i}(\mathbf{x})$ , $i=1,\dots,k$ be multivariate polynomials in $\mathbf{x}=(x_{1},x_{2},\dots,x_{m})^{\top}\in\mathbb{R}^{m}$ , i.e., $p(\mathbf{x}),g_{i}(\mathbf{x})\in\mathbb{R}[\mathbf{x}]$  [4]. The polynomial optimization problem (POP) can be stated as follows:

Since polynomial equalities can be expressed using pairs of opposite inequalities, i.e., $$g_{i}(\mathbf{x})=0\,\,\Leftrightarrow\,\, g_{i}(\mathbf{x})\geq0\wedge g_{i}(\mathbf{x})\leq0,$$ POP encompasses all polynomial optimization problems. In general, POP is an NP-hard problem. In [6], Lasserre suggested to convexify POP using a hierarchy of semidefinite (SDP) relaxations $\mathcal{P}_{\delta}$ , $\delta=1,2,\dots$ Since the SDP problems $\mathcal{P}_{\delta}$ take the form of linear matrix inequalities (LMI), the hierarchy is sometimes called Lasserre’s LMI hierarchy and $\mathcal{P}_{\delta}$ is called a LMI relaxation of order $\delta$ . The most agreeable property of Lasserre’s hierarchy is the fact that the minima of $\mathcal{P}_{\delta}$ form a monotonically non-decreasing sequence of the lower bounds on the global minimum of Problem 1↑. Moreover, in most cases a relaxation $\mathcal{P}_{\delta'}$ , $\delta'\in\mathbb{N}$ exists such that its minimum is equal to the global minimum of Problem 1↑.

Here, $\mathbb{S}^{n_{i}}(\mathbb{R}[\mathbf{x}])$ stands for the set of $n_{i}\times n_{i}$ symmetric matrices with polynomial entries and the condition $\mathtt{G}_{i}(\mathbf{x})\succeq0$ means that the polynomial matrix $\mathtt{G}_{i}(\mathbf{x})$ is positive semidefinite. Note that if $n_{i}=1$ , $i=1,...,k$ , Problem 2↑ becomes equivalent to Problem 1↑, i.e., POP is a subset of PMI. Also note that if the polynomial degree of $p(\mathbf{x})$ and the maximal polynomial degree of all the entries of $\mathtt{G}_{i}(\mathbf{x})$ , $i=1,...,k$ , is one, PMI becomes LMI. Even though PMI is a strict superset of POP, it was shown in [5] that PMI is amendable to solution using an LMI hierarchy analogous to the LMI hierarchy for POP. GpoSolver implements solutions for PMI, POP, and LMI problems. We refer to these problems by the name of the largest problem set as PMI problems.

Mathematically, PMI classes lead to problems that are equivalent to Problem 2↑. However, not every aspect of Problem 2↑ can be parametrized in GpoSolver at the moment, e.g., the number of constraints must be fixed. The precise formulation of PMI classes solvable by GpoSolver is stated by the following problem:

This parametrization allows for two sets of parameters $\boldsymbol{\alpha}\in\mathbb{R}^{m'}$ , $\boldsymbol{\beta}_{j}\in\mathbb{R}^{m''}$ , $j=1,\dots,\ell$ and for a summation-based cost function. We call the parameters $\boldsymbol{\alpha}$ problem parameters and they can appear in the cost function as well as in the constraints. The values of these parameters are to be specified during the problem solving phase, however, the number of problem parameters $m'$ needs to be specified in the modeling phase. The second set of parameters $\boldsymbol{\beta}_{j}\in\mathbb{R}^{m''}$ , $j=1,\dots,\ell$ we call residual parameters. Again, the number of parameters $m''$ must be specified in the modeling phase. The values of the residual parameters as well as the number of the parameter blocks $\ell$ are instance specific. The instance-specific parameter $\ell$ together with the summation-based cost function allow for PMI classes based on minimization of residual based cost functions. In the modeling phase, only the form of $r(\boldsymbol{\alpha},\boldsymbol{\beta}_{j},\mathbf{x})$ must be specified. If the problem contains no residual parameters, then $\ell=1$ by default. To sum it up, the values of $k$ , $m$ , $m'$ , $m''$ and $n_{i}$ need to be specified in the modeling phase. The parameter values as well as the number of residual blocks $\ell$ are instance dependent and are to be specified in the solving phase.