I am reading the paper "OPTIMAL INEQUALITIES IN PROBABILITY THEORY: A CONVEX OPTIMIZATION APPROACH" by BERTSIMAS and POPESCU. In the paper, the authors derived a duality problem for an infinite-dimensional optimization problem. I am not sure how to derive the following:

The primal is:

$\max_\mu \quad \int_S \textbf{1}d\mu$

subject to $\int_\Omega \bar z^kd\mu=\sigma_\kappa$, $\forall \kappa\in J_k$.

The dual is:

$\min_{y\in \mathcal{R}^{|J_k|}} \quad \sum_{\kappa\in J_k}y_\kappa \sigma_\kappa$

subject to $g(\bar z)=\sum_{\kappa\in J_k}y_\kappa\bar z^\kappa\geq 1$, $\forall \bar z \in S$.

In the above, $\mu$ is a probability measure and $S\subseteq \Omega\subseteq\mathcal{R}^n$. Could anyone tell me how to deal with this kind of infinite-dimensional optimization problems?

I am reading the paper "OPTIMAL INEQUALITIES IN PROBABILITY THEORY: A CONVEX OPTIMIZATION APPROACH" by BERTSIMAS and POPESCU. In the paper, the authors derived a duality problem for an infinite-dimensional optimization problem. I am not sure how to derive the following:

The primal is:

$\max_\mu \quad \int_S \textbf{1}d\mu$

subject to $\int_\Omega \bar z^kd\mu=\sigma_\kappa$, $\forall \kappa\in J_k$.

The dual is:

$\min_{y\in \mathcal{R}^{|J_k|}} \quad \sum_{\kappa\in J_k}y_\kappa \sigma_\kappa$

subject to $g(\bar z)=\sum_{\kappa\in J_k}y_\kappa\bar z^\kappa\geq 1$, $\forall \bar z \in S$, and $g(\bar z)=\sum_{\kappa\in J_k}y_{\kappa}\bar z^\kappa\geq 0$, $\forall \bar z\in\Omega$.

In the above, $\mu$ is a probability measure and $S\subseteq \Omega\subseteq\mathcal{R}^n$. Moreover, $\bar z=(z_1,\ldots,z_n)'$, $\kappa=(k_1,\ldots,k_n)'$, $\bar z^\kappa=z_1^{k_1}\cdots z_n^{k_n}$, and $$ J_k=\{ \kappa=(k_1,\ldots,k_n)'|k_1+\cdots+k_n\leq k,~k_j\in\mathcal{Z}_+,~j=1,\ldots,n \}. $$ Could anyone tell me how to deal with this kind of infinite-dimensional optimization problem?