# Inverse multiobjective optimization: Inferring decision criteria from data

## TL;DR

This paper introduces a method to infer the objectives of a multiobjective optimization problem from data points, enabling identification of decision criteria and construction of surrogate models efficiently.

## Contribution

It presents a novel inverse optimization approach that constructs objectives from data, solving a linear system to identify Pareto critical sets and enabling applications in surrogate modeling.

## Key findings

- Method effectively identifies objectives from noisy data.
- Approach accelerates solving expensive multiobjective problems.
- Demonstrated success on multiple examples.

## Abstract

It is a very challenging task to identify the objectives on which a certain decision was based, in particular if several, potentially conflicting criteria are equally important and a continuous set of optimal compromise decisions exists. This task can be understood as the inverse problem of multiobjective optimization, where the goal is to find the objective vector of a given Pareto set. To this end, we present a method to construct the objective vector of a multiobjective optimization problem (MOP) such that the Pareto critical set contains a given set of data points or decision vectors. The key idea is to consider the objective vector in the multiobjective KKT conditions as variable and then search for the objectives that minimize the Euclidean norm of the resulting system of equations. By expressing the objectives in a finite-dimensional basis, we transform this problem into a homogeneous, linear system of equations that can be solved efficiently. There are many important potential applications of this approach. Besides the identification of objectives (both from clean and noisy data), the method can be used for the construction of surrogate models for expensive MOPs, which yields significant speed-ups. Both applications are illustrated using several examples.