Inverse optimization problems with multiple weight functions

TL;DR

This paper introduces a new class of inverse optimization problems involving multiple weight functions, providing LP duality-based characterizations, complexity insights, and conditions for solutions that modify only the input solution's elements.

Contribution

It extends inverse optimization to multiple weights, offers min-max LP characterizations, and analyzes the complexity and solution structure of these problems.

Findings

Optimal deviation vectors may be non-integral even with integral weights.

LP duality yields min-max characterizations for the deviation norm.

Conditions are provided for deviations that only modify the input solution.

Abstract

We introduce a new class of inverse optimization problems in which an input solution is given together with $k$ linear weight functions, and the goal is to modify the weights by the same deviation vector $p$ so that the input solution becomes optimal with respect to each of them, while minimizing $\|p\|_1$. In particular, we concentrate on three problems with multiple weight functions: the inverse shortest $s$-$t$ path, the inverse bipartite perfect matching, and the inverse arborescence problems. Using LP duality, we give min-max characterizations for the $\ell_1$-norm of an optimal deviation vector. Furthermore, we show that the optimal $p$ is not necessarily integral even when the weight functions are so, therefore computing an optimal solution is significantly more difficult than for the single-weighted case. We also give a necessary and sufficient condition for the existence of an optimal deviation vector that changes the values only on the elements of the input solution, thus giving a unified understanding of previous results on arborescences and matchings.