The Discrete Moment Problem with Nonconvex Shape Constraints.

The discrete moment problem aims to find a worst-case discrete distribution that satisfies a given set of moments. This paper studies the discrete moment problems with additional shape constraints that guarantee the worst-case distribution is either log-concave (LC) or has an increasing failure rate (IFR) or increasing generalized failure rate (IGFR). These classes are useful in practice, with applications in revenue management, reliability, and inventory control. The authors characterize the structure of optimal extreme point distributions and show, for example, that an optimal extreme point solution to a moment problem with m moments and LC shape constraints is piecewise geometric with at most m pieces. Using this optimality structure, they design an exact algorithm for computing optimal solutions in a low-dimensional space of parameters. The authors leverage this structure to study a robust newsvendor problem with shape constraints and compute optimal solutions. The discrete moment problem is a foundational problem in distribution-free robust optimization, where the goal is to find a worst-case distribution that satisfies a given set of moments. This paper studies the discrete moment problems with additional shape constraints that guarantee the worst-case distribution is either log-concave (LC), has an increasing failure rate (IFR), or increasing generalized failure rate (IGFR). These classes of shape constraints have not previously been studied in the literature, in part due to their inherent nonconvexities. Nonetheless, these classes are useful in practice, with applications in revenue management, reliability, and inventory control. We characterize the structure of optimal extreme point distributions under these constraints. We show, for example, that an optimal extreme point solution to a moment problem with m moments and LC shape constraints is piecewise geometric with at most m pieces. This optimality structure allows us to design an exact algorithm for computing optimal solutions in a low-dimensional space of parameters. We also leverage this structure to study a robust newsvendor problem with shape constraints and compute optimal solutions. [ABSTRACT FROM AUTHOR]