(328d) Data-Driven Stochastic Robust Optimization: General Modeling Framework and Efficient Computational Algorithm for Handling Labeled Multi-Class Uncertainty Data

Ning, C., Cornell University
You, F., Cornell University
Optimization under uncertainty has attracted tremendous attention from both academia and industry [1]. A plethora of mathematical programming techniques, such as stochastic programming [2], and robust optimization [3], have been proposed. These techniques have their respective strengths and weaknesses, which leads to different scopes of applications [4-7]. Stochastic programming is an approach that focuses on the expected performance of a solution by leveraging probability distributions. However, it becomes computationally intractable as the number of scenarios increases [8]. Robust optimization provides an alternative paradigm that does not require an accurate knowledge of probability distributions, and offers an attractive benefit of computational tractability. Nevertheless, the robust optimization solution usually suffers from the conservatism issue. The state-of-the-art approaches leverage the synergy of different optimization methods to inherit their corresponding advantages and complement respective drawbacks. However, these approaches fall short of leveraging uncertainty data to benefit decision making processes. Thanks to massive amounts of data and great advancements in machine learning, data-driven optimization emerges as a promising paradigm [9, 10].

Uncertainty data in large data sets are often collected from various conditions, which are encoded by class labels. By leveraging the label information, we propose a novel data-driven stochastic robust optimization (DDSRO) framework that systematically and automatically handles the labeled multi-class uncertainty data. To account for the uncertainty data with different labels, we develop a modeling framework that leverages the synergy of stochastic programming and robust optimization. Stochastic programming is nested in the outer problem to optimize the expected objective over categorical classes. Operational decisions need to hedge against the worst case to ensure the robustness of operations. Besides, estimating a joint probability distribution of a high-dimensional uncertainty is much more challenging than estimating a categorical distribution. Therefore, robust optimization, which does not require an accurate knowledge of probability distribution, is nested in the inner problem. A bag of Dirichlet process mixture models is employed to construct uncertainty sets from the labeled multi-class uncertainty data [11, 12]. This data-driven nonparametric uncertainty set could adjust its complexity to that of data automatically, thus truthfully capturing the nature of uncertainty within the same class. The state-of-the-art machine learning technique is seamlessly integrated with the stochastic robust optimization framework in the form of a multi-level mixed-integer linear program (MILP). We further develop a tailored column-and-constraint generation (C&CG) algorithm [13] to solve the DDSRO problem efficiently. It is worth noting that the proposed DDSRO framework is an extension of data-driven adaptive nested robust optimization (DDANRO) [9], which was recently proposed to handle uncertainty data homogeneously without considering the label information. DDSRO greatly enhances the capabilities of DDANRO, and expands its scope of applications to more complex data structures. To demonstrate the advantages of the proposed framework and the efficiency of the computational algorithm, a motivating numerical example and two case studies on strategic planning of process networks [14] under supply and demand uncertainties are presented.


[1] I. E. Grossmann, R. M. Apap, B. A. Calfa, P. García-Herreros, and Q. Zhang, "Recent advances in mathematical programming techniques for the optimization of process systems under uncertainty," Computers & Chemical Engineering, vol. 91, pp. 3-14, 2016.

[2] J. R. Birge and F. Louveaux, Introduction to stochastic programming: Springer Science & Business Media, 2011.

[3] A. Ben-Tal, L. E. Ghaoui, and A. Nemirovski, Robust Optimization: Princeton University Press, 2009.

[4] H. Shi and F. You, "A computational framework and solution algorithms for two-stage adaptive robust scheduling of batch manufacturing processes under uncertainty," AIChE Journal, vol. 62, pp. 687-703, 2016.

[5] J. Gong, D. J. Garcia, and F. You, "Unraveling optimal biomass processing routes from bioconversion product and process networks under uncertainty: An adaptive robust optimization approach," ACS Sustainable Chemistry & Engineering, vol. 4, pp. 3160-3173, 2016.

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

[7] J. Gong and F. You, "Optimal processing network design under uncertainty for producing fuels and value-added bioproducts from microalgae: Two-stage adaptive robust mixed integer fractional programming model and computationally efficient solution algorithm," AIChE Journal, vol. 63, pp. 582-600, 2017.

[8] B. H. Gebreslassie, Y. Yao, and F. You, "Design under uncertainty of hydrocarbon biorefinery supply chains: Multiobjective stochastic programming models, decomposition algorithm, and a Comparison between CVaR and downside risk," AIChE Journal, vol. 58, pp. 2155-2179, 2012.

[9] C. Ning and F. You, "Data-driven adaptive nested robust optimization: General modeling framework and efficient computational algorithm for decision making under uncertainty," AIChE Journal, 2017. DOI: 10.1002/aic.15717

[10] D. Bertsimas, V. Gupta, and N. Kallus, "Data-driven robust optimization," Mathematical Programming, 2017. DOI:10.1007/s10107-017-1125-8

[11] T. Campbell and J. P. How, "Bayesian nonparametric set construction for robust optimization," in 2015 American Control Conference (ACC), 2015, pp. 4216-4221.

[12] D. M. Blei and M. I. Jordan, "Variational inference for Dirichlet process mixtures," Bayesian analysis, vol. 1, pp. 121-143, 2006.

[13] B. Zeng and L. Zhao, "Solving two-stage robust optimization problems using a column-and-constraint generation method," Operations Research Letters, vol. 41, pp. 457-461, 2013.

[14] F. You and I. E. Grossmann, "Stochastic inventory management for tactical process planning under uncertainties: MINLP models and algorithms," AIChE Journal, vol. 57, pp. 1250-1277, 2011.