We consider a residuals-based distributionally robust optimization (DRO) model, where the underlying uncertainty depends on both covariate information and our decisions. We adopt both parametric and nonparametric regression models to learn the latent decision dependency and construct a nominal distribution (thereby ambiguity sets) around the learned model using empirical residuals from the regressions. We formulate the ambiguity set via the Wasserstein distance, where the nominal distribution is both decision- and covariate-dependent. We provide conditions under which desired statistical properties such as asymptotic optimality, rate of convergence, and finite sample guarantees are satisfied. To solve the resulting DRO model, we develop a specialized Bender's decomposition algorithm with nonlinear cuts and prove its finite convergence. Through numerical experiments, we illustrate the effectiveness of our approach and the benefits of integrating decision dependency into a residuals-based DRO framework.
The stochastic variational inequality problem (SVIP) is an equilibrium model that includes random variables and has been widely applied in various fields such as economics and engineering. Expected residual minimization (ERM) is an established model for obtaining a reasonable solution for the SVIP, and its objective function is an expected value of a suitable merit function for the SVIP. However, the ERM is restricted to the case where the distribution is known in advance. We extend the ERM to ensure the attainment of robust solutions for the SVIP under the uncertainty distribution (the extend…
In this paper, we develop a distributionally robust chance-constrained formulation of the Optimal Power Flow problem (OPF) whereby the system operator can leverage contextual information. For this purpose, we exploit an ambiguity set based on probability trimmings and optimal transport through which the dispatch solution is protected against the incomplete knowledge of the relationship between the OPF uncertainties and the context that is conveyed by a sample of their joint probability distribution. We provide a tractable reformulation of the proposed distributionally robust chance-constrained…
We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of regression setups and DRO ambiguity sets. We investigate asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through…
In this paper, we develop a distributionally robust model predictive control framework for the control of wind farms with the goal of power tracking and mechanical stress reduction of the individual wind turbines. We introduce an ARMA model to predict the turbulent wind speed, where we merely assume that the residuals are sub-Gaussian noise with statistics contained in an moment-based ambiguity set. We employ a recently developed distributionally model predictive control scheme to ensure constraint satisfaction and recursive feasibility of the control algorithm. The effectiveness of the approa…
We explore generalizations of some integrated learning and optimization frameworks for data-driven contextual stochastic optimization that can adapt to heteroscedasticity. We identify conditions on the stochastic program, data generation process, and the prediction setup under which these generalizations possess asymptotic and finite sample guarantees for a class of stochastic programs, including two-stage stochastic mixed-integer programs with continuous recourse. We verify that our assumptions hold for popular parametric and nonparametric regression methods.
This paper presents a novel methodology for tractably solving optimal control and offline reinforcement learning problems for high-dimensional systems. This work is motivated by the ongoing challenges of safety, computation, and optimality in high-dimensional optimal control. We address these key questions with the following approach. First, we identify a sequence-modeling surrogate methodology which takes as input the initial state and a time series of control inputs, and outputs an approximation of the objective function and trajectories of constraint functions. Importantly this approach ent…
Desautels Faculty of Management（德索特尔管理学院）；McGill University（麦吉尔大学）；GERAD（GERAD（蒙特利尔决策科学、运筹学和统计学研究组））；HEC Montréal（蒙特利尔高等商学院）；University of Cyprus（塞浦路斯大学）
Contextual Optimization under Covariate Shift: A Robust Approach by Intersecting Wasserstein Balls
Department of Industrial Engineering and Operations Research, Columbia University（哥伦比亚大学工业工程与运营管理系）；Department of Management, University of Toronto Mississauga, Rotman School of Management, University of Toronto（多伦多大学滑铁卢分校管理系，罗特曼管理学院，多伦多大学）；School of Economics and Management, Tsinghua University（清华大学经济管理学院）
