Outer approximating coherent lower probabilities with belief functions

From an epistemic point of view, coherent lower probabilities allow us to model the imprecise information about a partially unknown probability. However, there are some issues that hinder their use in practice. Since belief functions are easier to deal with, we propose to approximate the coherent lower probability by a belief function that is at the same time as close as possible to the initial coherent lower probability while not including additional information. We show that this problem can be tackled by means of linear programming, and investigate the features of the set of optimal solutions. Moreover, we emphasize the dfferences with the outer approximations by 2-monotone lower probabilities. We also study the problem for two particular cases of belief functions that are computationally easier to handle: necessity measures and probability boxes.