(393b) Data-Driven Stochastic Robust Optimization for Process Operation Under Uncertainty

Optimization under uncertainty has attracted tremendous attention from both academia and industry (Grossmann et al., 2016). Uncertain parameters, if not being accounted for, could render the solution of an optimization problem suboptimal or even infeasible. To this end, a plethora of mathematical programming techniques have been proposed. These techniques have their respective strengths and weaknesses, which lead to different application scopes. Stochastic programming focuses on the expected performance of a solution by leveraging the scenarios of uncertainty realization and their probability distribution. However, this approach requires accurate information on the probability distribution, and the resulting optimization problem could become computationally challenging. Robust optimization provides an alternative paradigm that does not require accurate knowledge on probability distributions. The state-of-the-art approaches for optimization under uncertainty leverage the synergy of different optimization methods to inherit their corresponding strengths and complement respective weaknesses (Yue and You, 2016).

Labeled multi-class data are ubiquitous in a variety of areas and disciplines. For example, process data are labeled with the operating modes of chemical plants (Afzal et al., 2017). Due to the massive amount of available uncertainty data and dramatic progress in big data analytics, data-driven optimization under uncertainty emerges as a promising paradigm (Shang et al., 2017). Most existing data-driven optimization methods are restricted to unlabeled uncertainty data. Discarding labels and applying the existing methods to labeled uncertainty data ignores information embedded in labels, possibly compromising results and leading to sub-optimal solutions. Consequently, new modeling frameworks and computational algorithms are required for handling labeled multi-class uncertainty data.

This paper proposes a data-driven stochastic robust optimization (DDSRO) framework that systematically and automatically handles labeled multi-class uncertainty data. In large datasets, uncertainty data are often attached with labels to indicate their data classes. For the labeled multi-class uncertainty data, we propose a data-driven uncertainty modeling process that includes two parts, i.e. probability distribution estimation for different data classes, and uncertainty set construction. The probability distribution for data classes is learned from labeled uncertainty data through maximum likelihood estimation for the multinoulli distribution. To capture the nature of uncertainty data with different labels, a group of Dirichlet process mixture models is employed to model uncertainty with a variational inference algorithm (Ning and You, 2017). These two pieces of uncertainty information are incorporated into the DDSRO framework through a bi-level optimization structure. Two-stage stochastic programming is nested in the outer problem to optimize the expected objective over categorical data classes; robust optimization is nested as the inner problem to hedge against high-dimensional uncertainty and ensure computational tractability. To demonstrate effectiveness of the proposed method, process network planning application (You and Grossmann, 2011) is presented.
References

M. Afzal, W. Tan, T. Chen, 2017, Process Monitoring for Multimodal Processes With Mode-Reachability Constraints. IEEE Transactions on Industrial Electronics, 64, 4325-4335.

I. Grossmann, R. Apap, B. Calfa, P. García-Herreros, Q. Zhang, 2016, Recent advances in mathematical programming techniques for the optimization of process systems under uncertainty, Computers & Chemical Engineering, 91, 3-14.

C. Ning, F. You, 2017, Data‐driven adaptive nested robust optimization: General modeling framework and efficient computational algorithm for decision making under uncertainty. AIChE Journal, 63, 3790-3817.

C. Shang, X. Huang, F. You, 2017, Data-driven robust optimization based on kernel learning. Computers & Chemical Engineering, 106, 464-479.

F. You, I. Grossmann, 2011, Stochastic inventory management for tactical process planning under uncertainties: MINLP models and algorithms. AIChE Journal, 57, 1250-1277.

D. Yue, F. You, 2016, Optimal supply chain design and operations under multi‐scale uncertainties: Nested stochastic robust optimization modeling framework and solution algorithm. AIChE Journal, 62, 3041-3055.