# Newton-type algorithms for inverse optimization II: weighted span objective

## TL;DR

This paper introduces a new weighted span objective for inverse optimization, providing a min-max characterization and a polynomial-time Newton-type algorithm for finding optimal deviations, aiming for balanced cost modifications.

## Contribution

It proposes the weighted span as a novel objective in inverse optimization and develops an efficient algorithm for its computation.

## Key findings

- Min-max characterization of the weighted span
- A strongly polynomial Newton-type algorithm for unit weights
- Balanced cost modifications in inverse optimization

## Abstract

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 give a min-max characterization for the minimum weighted span of a feasible deviation vector, and provide a Newton-type algorithm for finding one that runs in strongly polynomial time in the case of unit weights.