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.

识别某个决策所依据的目标是一项艰巨的任务，尤其是在多个潜在冲突的标准同等重要并且存在一套连续的最佳折衷决策的情况下。这项任务可以理解为多目标优化的反问题，其目标是找到给定Pareto集的目标函数向量。为此，我们提出了一种构造无约束多目标优化问题（MOP）的目标函数向量的方法，以使Pareto临界集包含具有给定的KKT乘数的给定数据点集。如果找不到这样的MOP，则该方法将生成一个MOP，其Pareto关键集合至少与数据点接近。关键思想是将多目标KKT条件下的目标函数向量视为变量，然后搜索使所得方程组的欧几里得范数最小的目标。通过在有限维的基础上表达目标，我们将此问题转化为可以有效求解的齐次线性方程组。这种方法的潜在应用包括确定目标（从干净的和嘈杂的数据中）以及为昂贵的MOP构建替代模型。