Inverse multiobjective optimization: Inferring decision criteria from data

Abstract

It is a 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 function vector of a given Pareto set. To this end, we present a method to construct the objective function vector of an unconstrained multiobjective optimization problem (MOP) such that the Pareto critical set contains a given set of data points with prescribed KKT multipliers. If such an MOP can not be found, then the method instead produces an MOP whose Pareto critical set is at least close to the data points. The key idea is to consider the objective function 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. Potential applications of this approach include the identification of objectives (both from clean and noisy data) and the construction of surrogate models for expensive MOPs.

Inverse attribute‐based optimization with an application in assortment optimization

“…Inverse multiobjective optimization has focused on imputing the weights of subobjectives under different assumptions on Pareto optimality or feasibility of observations and availability of a prior weight vector (Roland et al., 2013; Chan et al., 2014; Chan and Lee, 2018; Naghavi et al., 2019; Ajayi et al., 2020; Dong and Zeng, 2020; Gebken and Peitz, 2021). If an attribute‐based forward problem can be reformulated as a multiobjective problem, then, under appropriate assumptions, the methods of inverse multiobjective optimization can be applied to solving IAO problems as well.…”