[Paper Review] The discrete moment problem with nonconvex shape constraints

Xi Chen, Simai He+3 more|arXiv (Cornell University)|Aug 7, 2017

Risk and Portfolio Optimization39 references3 citations

TL;DR

This paper develops an exact algorithm for solving discrete moment problems under nonconvex shape constraints—specifically log-concave and increasing failure rate (IFR) distributions—by leveraging reverse convex optimization. It proves that optimal extreme-point solutions are piecewise geometric with at most m pieces for m moments, enabling efficient computation in low-dimensional parameter spaces and yielding tighter bounds than unimodal relaxations.

ABSTRACT

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 or has an increasing failure rate. 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. We characterize the structure of optimal extreme point distributions by developing new results in reverse convex optimization, a lesser-known tool previously employed in designing global optimization algorithms. We are able to show, for example, that an optimal extreme point solution to a moment problem with $m$ moments and log-concave shape constraints is piecewise geometric with at most $m$ pieces. Moreover, this structure allows us to design an exact algorithm for computing optimal solutions in a low-dimensional space of parameters. Moreover, We describe a computational approach to solving these low-dimensional problems, including numerical results for a representative set of instances.

Motivation & Objective

• To address the lack of theoretical and computational frameworks for discrete moment problems with nonconvex shape constraints such as log-concavity and IFR.
• To characterize the structure of optimal extreme-point distributions under these nonconvex constraints.
• To develop an exact computational algorithm for solving these problems in low-dimensional parameter spaces.
• To demonstrate through numerical results that log-concave constraints yield tighter bounds than unimodal relaxations.
• To extend the applicability of moment problem solutions to practical robust optimization settings where log-concave or IFR distributions are common.

Proposed method

• Uses reverse convex optimization, a lesser-known tool in global optimization, to analyze and structure optimal solutions under nonconvex shape constraints.
• Derives analytical conditions for optimality by transforming the problem into a system of nonlinear equations involving derivatives of implicit functions.
• Applies Newton’s method on a one-dimensional interval to solve for roots of the optimality conditions, ensuring convergence to global optima.
• Imposes the constraint α + β ≤ 0 to enforce log-concavity on the middle index of the distribution, preserving the shape property.
• Reduces the original high-dimensional problem to a low-dimensional parametric search over critical parameters (α, β), enabling exact solution computation.
• Employs a parametric formulation where the optimal solution is determined by solving a system of equations derived from KKT conditions and implicit function derivatives.

Experimental results

Research questions

• RQ1 What is the structural form of optimal extreme-point solutions to discrete moment problems with nonconvex log-concave shape constraints?
• RQ2 How can reverse convex optimization be applied to characterize global optima in nonconvex distributional constraints?
• RQ3 Can tighter bounds on tail probabilities be achieved by enforcing log-concavity versus unimodality in discrete moment problems?
• RQ4 What is the computational complexity and feasibility of solving such problems exactly in low-dimensional parameter spaces?
• RQ5 To what extent do log-concave and IFR constraints improve robust optimization outcomes compared to moment-only or unimodal constraints?

Key findings

• An optimal extreme-point solution to a discrete moment problem with m moments and log-concave constraints is piecewise geometric with at most m pieces.
• The proposed algorithm computes exact solutions by reducing the problem to a one-dimensional search over critical parameters, enabling global optimization.
• Numerical results show that log-concave constraints yield significantly tighter upper and lower bounds on tail probabilities than unimodal relaxations.
• For n=5, S₁=1.9, S₂=1.3, the log-concave bound is 0.9000–1.0000, compared to 0.8750–1.0000 under unimodal relaxation.
• For n=11, S₁=5.2, S₂=15.1, the log-concave bound is 0.9310–0.9921, while the unimodal bound is looser at 0.9208–1.0000.
• The method achieves tighter bounds across all tested instances, confirming the value of enforcing stronger shape constraints like log-concavity.