Envelope theorem and discontinuous optimisation: the case of positioning choice problems

TL;DR

This paper establishes that the value function in positioning choice problems is almost everywhere differentiable despite discontinuities, by developing an envelope theorem using Dini superdifferentials.

Contribution

It introduces an envelope theorem for positioning choice problems, showing differentiability properties and first-order conditions even with discontinuous objectives.

Findings

Value function is almost everywhere differentiable.

Dini superdifferential is well-defined for maxima.

Envelope theorem applies to positioning choice problems.

Abstract

This article examines differentiability properties of the value function of positioning choice problems, a class of optimisation problems in finite-dimensional Euclidean spaces. We show that positioning choice problems' value function is always almost everywhere differentiable even when the objective function is discontinuous. To obtain this result we first show that the Dini superdifferential is always well-defined for the maxima of positioning choice problems. This last property allows to state first-order necessary conditions in terms of Dini supergradients. We then prove our main result, which is an ad-hoc envelope theorem for positioning choice problems. Lastly, after discussing necessity of some key assumptions, we conjecture that similar theorems might hold in other spaces as well.