Discrete Math Seminar: Chris Ryan

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 an 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 of distributions are useful in practice with applications in revenue management, reliability, and inventory control. We characterize the structure of optimal extreme point distributions. 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 is joint work with Xi Chen (NYU, Stern School of Business), Simai He (Shanghai University of Finance and Economics, School of Information Management and Engineering), Bo Jiang (Shanghai University of Finance and Economics, School of Information Management and Engineering), and Teng Zhang (Stanford, Management Science and Engineering).