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.
Highlights
When applying optimization to real-world problems, there are often multiple quantities that have to be optimized at the same time
We address the task of finding the objective function vector whose extended Pareto critical set is as close to a given data set as possible
We present a way to construct an objective function vector of an multiobjective optimization problem (MOP) such that its extended Pareto critical set contains a given data set
Summary
When applying optimization to real-world problems, there are often multiple quantities that have to be optimized at the same time. Part of the literature in the single objective case is concerned with finding a weighting vector for the objectives of an MOP such that a given feasible point is Pareto optimal (cf [6,7]) This area is referred to as inverse multiobjective optimization, but differs from the context in this paper. In [12], a method was presented to find the parameters of a parameter-dependent, convex and constrained MOP such that its Pareto set contains a set of given, noisy points (modeled via probability distributions). If the KKT vectors are known (as in the applications presented in this paper), allowing nonconvex objectives will significantly increase the complexity of the geometry and topology of the data that can be handled This will be highlighted in our examples, where many of the objective functions are non-convex. For the computation of the extended Pareto critical sets in this article, we use the Continuation Method CONT-Recover from [34]
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