  Shape-constrained optimization arises in a wide range of problems including
distributionally robust optimization (DRO) that has surging popularity in
recent years. In the DRO literature, these problems are usually solved via
reduction into moment-constrained problems using the Choquet representation.
While powerful, such an approach could face tractability challenges arising
from the geometries and the compatibility between the shape and the objective
function and moment constraints. In this paper, we propose an alternative
methodology to solve shape-constrained optimization problems by integrating
sample average approximation with importance sampling, the latter used to
convert the distributional optimization into an optimization problem over the
likelihood ratio with respect to a sampling distribution. We demonstrate how
our approach, which relies on finite-dimensional linear programs, can handle a
range of shape-constrained problems beyond the reach of previous Choquet-based
reformulations, and entails vanishing and quantifiable optimality gaps.
Moreover, our theoretical analyses based on strong duality and empirical
processes reveal the critical role of shape constraints in guaranteeing
desirable consistency and convergence rates.
