Piecewise linear approximations of the standard normal first order loss function

TL;DR

This paper develops effective piecewise linear bounds for the standard normal first order loss function, enabling easier integration into optimization models and providing optimal parameters to minimize approximation errors.

Contribution

It introduces a comprehensive approach for piecewise linear approximation of the normal first order loss function, including optimal parameter computation for minimal error.

Findings

Provides linear bounds that can be embedded in MILP models.

Develops methods to compute optimal linearisation parameters.

Extends results to symmetric and normal distributions.

Abstract

The first order loss function and its complementary function are extensively used in practical settings. When the random variable of interest is normally distributed, the first order loss function can be easily expressed in terms of the standard normal cumulative distribution and probability density function. However, the standard normal cumulative distribution does not admit a closed form solution and cannot be easily linearised. Several works in the literature discuss approximations for either the standard normal cumulative distribution or the first order loss function and their inverse. However, a comprehensive study on piecewise linear upper and lower bounds for the first order loss function is still missing. In this work, we initially summarise a number of distribution independent results for the first order loss function and its complementary function. We then extend this discussion by focusing first on random variable featuring a symmetric distribution, and then on normally distributed random variables. For the latter, we develop effective piecewise linear upper and lower bounds that can be immediately embedded in MILP models. These linearisations rely on constant parameters that are independent of the mean and standard deviation of the normal distribution of interest. We finally discuss how to compute optimal linearisation parameters that minimise the maximum approximation error.