  We present and analyze a central cutting surface algorithm for general
semi-infinite convex optimization problems, and use it to develop a novel
algorithm for distributionally robust optimization problems in which the
uncertainty set consists of probability distributions with given bounds on
their moments. Moments of arbitrary order, as well as non-polynomial moments
can be included in the formulation. We show that this gives rise to a hierarchy
of optimization problems with decreasing levels of risk-aversion, with classic
robust optimization at one end of the spectrum, and stochastic programming at
the other. Although our primary motivation is to solve distributionally robust
optimization problems with moment uncertainty, the cutting surface method for
general semi-infinite convex programs is also of independent interest. The
proposed method is applicable to problems with non-differentiable semi-infinite
constraints indexed by an infinite-dimensional index set. Examples comparing
the cutting surface algorithm to the central cutting plane algorithm of
Kortanek and No demonstrate the potential of our algorithm even in the solution
of traditional semi-infinite convex programming problems whose constraints are
differentiable and are indexed by an index set of low dimension. After the rate
of convergence analysis of the cutting surface algorithm, we extend the
authors' moment matching scenario generation algorithm to a probabilistic
algorithm that finds optimal probability distributions subject to moment
constraints. The combination of this distribution optimization method and the
central cutting surface algorithm yields a solution to a family of
distributionally robust optimization problems that are considerably more
general than the ones proposed to date.
