Inverse multiobjective optimization: Inferring decision criteria from data

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.

Imputing radiobiological parameters of the linear-quadratic dose-response model from a radiotherapy fractionation plan

“…This paper proposes an inverse optimization Tarantola (1987) approach to study this question. The mathematical community has worked on inverse optimization in shortest path problems Toint 1992, Burton andToint 1994), finite-dimensional LPs (Ahuja and Orlin 2001, Chan and Neal Kaw 2019, Tavaslioglu et al 2018 and conic programs (Iyengar and Kang 2005), combinatorial optimization (Heuberger 2004), network optimization Zhang 1995, Yang et al 1997), multi-objective optimization (Chan et al 2014, Chan and Lee 2018, Gebken and Peitz 2019, Naghavi et al 2019, Roland et al 2013, integer programs (Schaefer 2009), mixed-integer programs (Wang 2009), separable convex programs with linear constraints (Zhang and Xu 2010), polynomial optimization (Lasserre 2013), machine learning/statistics , Tan et al 2018, countably infinite LPs (Ghate 2015), and minimum cost flow problems on countably infinite networks (Nourollahi and Ghate 2019). Inverse optimization has been applied to problems in geophysics (Neumann-Denzau and Behrens 1984, Nolet 1987, Tarantola 1987, Woodhouse and Dziewonski 1984, transportation (Dial 1999, Dial 2000, demand management (Carr and Lovejoy 2000), auctions (Beil and Wein 2003), production planning (Troutt et al 2006), healthcare (Ayer 2015, Erkin et al 2010, and finance (Bertsimas et al 2012).…”