Wasserstein Distributionally Robust Inverse Multiobjective Optimization

Abstract

Inverse multiobjective optimization provides a general framework for the unsupervised learning task of inferring parameters of a multiobjective decision making problem (DMP), based on a set of observed decisions from the human experts. However, the performance of this framework relies critically on the selection of appropriate decision making structure, a set of observed decisions that are sufficient and of high qualities, and a parameter space that contains enough information about the DMP. To hedge against the uncertainties in the hypothetical DMP, the data, and the parameter space, we investigate in this paper the distributionally robust approach for inverse multiobjective optimization. Specifically, we leverage the Wasserstein metric to construct a ball centering at the empirical distribution of these decisions. We then formulate a Wasserstein distributionally robust inverse multiobjective optimization problem (WRO-IMOP) that minimizes a worst-case expected loss function, where the worst case is taken over all distributions in the Wasserstein ball. We show that the excess risk of the WRO-IMOP estimator has a sub-linear convergence rate. Furthermore, we propose a semi-infinite reformulations of the WRO-IMOP and develop a cutting-plane algorithm that converges to any δ-optimal solution in finite iterations. Finally, we demonstrate the effectiveness of our method on both a synthetic multiobjective quadratic program and a real world portfolio optimization problem.

Inverse Optimization: Theory and Applications

“…Using techniques from distributionally robust optimization [67], they show that for linear forward models, IOP-DD-DRO(ℓ ASO , P N , CVaR, ǫ) can be re-formulated as a large conic optimization problem. More recently, Dong and Zeng [63] build on the distributionally robust framework of Esfahani et al [68] for multi-objective forward optimization models MO((θ, φ)) reduces to the conventional convex forward model FOP-CVX(θ, φ) when considering only a single objective. Dong and Zeng [63] use the expected value risk function ρ P (•) = E P [•] and distance minimization loss ℓ D , meaning that their inverse problem is a multiobjective distributionally robust generalization of the Inverse Distance problem of Aswani et al [17].…”

“…More recently, Dong and Zeng [63] build on the distributionally robust framework of Esfahani et al [68] for multi-objective forward optimization models MO((θ, φ)) reduces to the conventional convex forward model FOP-CVX(θ, φ) when considering only a single objective. Dong and Zeng [63] use the expected value risk function ρ P (•) = E P [•] and distance minimization loss ℓ D , meaning that their inverse problem is a multiobjective distributionally robust generalization of the Inverse Distance problem of Aswani et al [17]. Since the nominal inverse problem of Aswani et al [17] itself is NP-hard, the distributionally robust formulation is even more difficult to solve.…”

“…Since the nominal inverse problem of Aswani et al [17] itself is NP-hard, the distributionally robust formulation is even more difficult to solve. Using duality, Dong and Zeng [63] show that the distributionally robust problem can be re-formulated to a semi-infinite optimization problem under certain conditions, in which case they propose using constraint generation to solve the problem.…”

Inverse Multiobjective Optimization Through Online Learning

“…Different from the majority of existing literature, [17,18,19] take another perspective to explain the so called "data inconsistency": decision makers are driven by multiple criteria, and different people have different preferences or weights over those criteria, which leads them to make a variety of responses or choices. Then, it can be anticipated that once we remove the variance caused by such multi-criteria decision making from data, their quality or consistency can be greatly improved.…”