  Generalized moment problems optimize functional expectation over a class of
distributions with generalized moment constraints, i.e., the function in the
moment can be any measurable function. These problems have recently attracted
growing interest due to their great flexibility in representing nonstandard
moment constraints, such as geometry-mean constraints, entropy constraints, and
exponential-type moment constraints. Despite the increasing research interest,
analytical solutions are mostly missing for these problems, and researchers
have to settle for nontight bounds or numerical approaches that are either
suboptimal or only applicable to some special cases. In addition, the
techniques used to develop closed-form solutions to the standard moment
problems are tailored for specific problem structures. In this paper, we
propose a framework that provides a unified treatment for any moment problem.
The key ingredient of the framework is a novel primal-dual optimality
condition. This optimality condition enables us to reduce the original infinite
dimensional problem to a nonlinear equation system with a finite number of
variables. In solving three specific moment problems, the framework
demonstrates a clear path for identifying the analytical solution if one is
available, otherwise, it produces semi-analytical solutions that lead to
efficient numerical algorithms. Finally, through numerical experiments, we
provide further evidence regarding the performance of the resulting algorithms
by solving a moment problem and a distributionally robust newsvendor problem.
