Inverse Optimization with Noisy Data

• 成果类型: Article
• 署名作者: Aswani, Anil; Shen, Zuo-Jun (Max); Siddiq, Auyon
• 署名单位: University of California System; University of California Berkeley; University of California System; University of California Berkeley; University of California System; University of California Berkeley; University of California System; University of California Los Angeles
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2017.1705
• 发表日期: 2018
• 页码: 870-892
• 关键词: programming problem; Incentive design; least-squares; MODEL; PERSPECTIVE; algorithms; bilevel

摘要

Inverse optimization refers to the inference of unknown parameters of an optimization problem based on knowledge of its optimal solutions. This paper considers inverse optimization in the setting where measurements of the optimal solutions of a convex optimization problem are corrupted by noise. We first provide a formulation for inverse optimization and prove it to be NP-hard. In contrast to existing methods, we show that the parameter estimates produced by our formulation are statistically consistent. Our approach involves combining a new duality-based reformulation for bilevel programs with a regularization scheme that smooths discontinuities in the formulation. Using epi-convergence theory, we show the regularization parameter can be adjusted to approximate the original mverse optimization problem to arbitrary accuracy, which we use to prove our consistency results. Next, we propose two solution algorithms based on our duality-based formulation. The first is an enumeration algorithm that is applicable to settings where the dimensionality of the parameter space is modest, and the second is a semiparametric approach that combines nonparametric statistics with a modified version of our formulation. These numerical algorithms are shown to maintain the statistical consistency of the underlying formulation. Finally, using both synthetic and real data, we demonstrate that our approach performs competitively when compared with existing heuristics.