Shape-Constrained Distributional Optimization via Importance-Weighted Sample Average Approximation

TL;DR

This paper introduces a novel approach combining importance sampling with sample average approximation to efficiently solve shape-constrained distributional optimization problems, overcoming limitations of traditional methods.

Contribution

It proposes a new methodology that uses finite-dimensional linear programs and importance sampling to handle complex shape constraints in distributional optimization.

Findings

Handles a broader class of shape-constrained problems than previous methods.

Provides theoretical guarantees with vanishing and quantifiable optimality gaps.

Ensures consistency and convergence rates through strong duality and empirical process analysis.

Abstract

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.