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Chaosheng Dong, Bo Zeng
We study a new unsupervised learning task of inferring objective functions or constraints of a multiobjective decision making model, based on a set of observed decisions. Specifically, we formulate such a learning problem as an inverse multiobjective optimization problem (IMOP) and propose its first sophisticated model with statistical guarantees. Then, we some fundamental connections between IMOP, K-means clustering and manifold learning. More precisely, we prove that every K-means clustering problem can be transformed equivalently into an IMOP, and every IMOP can be conversely interpreted as a constrained K-means clustering problem. In addition, we show that the Pareto optimal set is a piecewise continuous manifold with an intrinsic dimension of $ p-1 $ (where $ p $ is the number of objectives) under suitable conditions. Hence, IMOP can also be interpreted as a manifold learning problem. Leveraging these critical insights and connections, we propose two algorithms to solve IMOP through manifold learning and clustering. Numerical results confirm the effectiveness of our model and the computational efficacy of algorithms.