Newton-Type Algorithms for Inverse Optimization: Weighted Span Objective

In inverse optimization problems, the goal is to modify the costs in an underlying optimization problem in such a way that a given solution becomes optimal, while the difference between the new and the original cost functions, called the deviation vector, is minimized with respect to some objective function. The \(\ell _1\) - and \(\ell _{\infty }\) -norms are standard objectives used to measure the size of the deviation. Minimizing the \(\ell _1\) -norm is a natural way of keeping the total change of the cost function low, while the \(\ell _{\infty }\) -norm achieves the same goal coordinate-wise. Nevertheless, none of these objectives is suitable to provide a balanced or fair change of the costs. In this paper, we initiate the study of a new objective that measures the difference between the largest and the smallest weighted coordinates of the deviation vector, called the weighted span. We provide a Newton-type algorithm for finding one that runs in strongly polynomial time in the case of unit weights.