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A Chebyshev Inequality Based on Bounded Support and Mean Absolute Deviation

Ernst Roos(e.j.roos***at***tilburguniversity.edu)
Ruud Brekelmans(r.c.m.brekelmans***at***tilburguniversity.edu)
Dick Den Hertog(d.denhertog***at***tilburguniversity.edu)

Abstract: The Chebyshev inequality is one of the most well-known results in classical probability theory. This inequality provides an upper bound on the tail probability of a random variable based on the fi rst two moments of its distribution. While this upper bound is tight, it has been criticized for only being attained by pathological distributions that abuse the unboundedness of the underlying support and are not considered realistic in many applications. In this paper, we provide an alternative tight lower and upper bound on the tail probability given a bounded support, mean and mean absolute deviation of the random variable. This result allows us to fi nd convex reformulations of single and joint ambiguous chance constraints with right-hand side uncertainty, and left-hand side uncertainty with specifi c structural properties as well as safe approximations to a wider class of ambiguous chance constraints. Applications of such ambiguous chance constraints include radiation therapy and inventory control problems, both of which we illustrate numerically.

Keywords: Chebyshev inequality, probability bounds, distributionally robust optimization, chance constraints

Category 1: Robust Optimization

Category 2: Stochastic Programming


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Entry Submitted: 07/12/2019
Entry Accepted: 07/12/2019
Entry Last Modified: 07/12/2019

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