Generalized Moment Problems
The Generalized Moment Problem (GMP) as described in Lasserre (2009) is a linear optimization problem with countably many constraints in the space of measures. Formally, it takes the form
where ${\color{red}\mu_i}$ are unkown (finite, positive, Borel) measures, $\Gamma$ is a countble set, and ${\color{green}b^\gamma}$ are given numbers. In order to be able to compute approximations to this problem we require ${\color{blue}f_i}$, ${\color{blue}h_i^\gamma}$, and ${\color{blue}g_i^j}$ to be polynomials.
Under some conditions a measure ${\color{red}\mu}$ supported on some set ${\color{blue}K}\subseteq \mathbb{R}^n$ is uniquely determined by its sequence of moments, defined by
where $x^\alpha := \prod_i x_i^{\alpha_i}$. This is for example the case when ${\color{blue}K}$ is compact, or when the sequence ${\color{red}(z_\alpha)_\alpha}$ satisfies the socalled Carleman's condition. Therefore one is often interested in the sequences of moments, rather than the measure itself, motivating the name generalized MOMENT problem.
Example: Polynomial Optimization
Given a polynomial ${\color{blue}f}$ and a set ${\color{blue}K} = \{x\in\mathbb{R}^n | {\color{blue}g^1}(x)\geq 0,\ldots,{\color{blue}g^{m}}(x)\geq 0 \}$ one is interested in the global minimum
This problem has a GMP formulation in terms, where we optimize over one single measure ${\color{red}\mu}$ and $\#\Gamma=1$:
Note that instead of optimizing over ${\color{red}x}\in{\color{blue}K}$, the variable of the GMP is the measure ${\color{red}\mu}$, hence turning a non-linear problem into a linear one.
The equivalence of both problems can be seen as follows: First note, that for every $x\in{\color{blue}K}$, the Dirac measure $\delta_{x}$ is a probability measure supported in ${\color{blue}K}$ and hence feasible for the GMP formulation. Together with the equality ${\color{blue}f}(x) = \int {\color{blue}f}d\delta_x$ this shows that the optimal value of the GMP formulation is always smaller than the one of the polynomial optimization problem.
For the other inequality let ${\color{red}f^\ast}$ denote the global minimum of ${\color{blue}f}$ on ${\color{blue}K}$ and let ${\color{red}\mu^\ast}$ be the optimal measure for the GMP formulation. Then
showing that the optimal value of the GMP formulation is always bigger than the one of the polynomial optimization problem - and in combination with the first part, equivalence of both problems.
In MomentOpt.jl this GMP can be modeled as demonstrated in polopt in the example folder.