Pareto-optimality in linear regression (Q1892523)

We have considered the linear regression problem as a vector optimization problem, whose set \({\mathcal S}\) of Pareto-optimal solutions has been characterized as: (i) The set of optimal solutions to problems where the error measure is a strictly increasing function. (ii) The union of all the elementary convex sets which are bounded. Furthermore, \({\mathcal S}\) can be obtained by standard vector linear programming techniques. These properties enable us to say, although depending on the error measure used one can obtain different predictions at a point \(x\), the possible predictions are given by the points in a certain interval \(\Pi (x)\), which can be explicitly obtained. An interesting line of future research could be the study of reductions of the set of optimal parameters by reducing the family \({\mathcal F}\) of globalizing functions \(\varphi\) allowed (by imposing, say, that \(\varphi\) must be a nondecreasing, symmetric and convex function, or a norm).