(186d) Data-Driven Robust Optimization with Principal Component Analysis and Kernel Smoothing

Authors: 
Ning, C., Cornell University
You, F., Cornell University
Optimization applications abound in many areas of science and engineering. In real practice, some parameters involved in optimization problems are subject to uncertainty [1]. The issue of uncertainty could render the solution of a deterministic optimization problem suboptimal or even infeasible. Therefore, optimization under uncertainty has attracted tremendous attention from both academia and industry. A wide array of mathematical programming techniques, such as stochastic programming and robust optimization, have been proposed [2]. The key idea of stochastic programming is to model the randomness in uncertain parameters with probability distributions. However, this approach requires accurate information on probability distribution. Furthermore, the resulting optimization problem is usually computationally challenging. By introducing “wait-and-see” decisions, adaptive robust optimization (ARO) approach well models the sequential nature of decision-making processes, and ameliorates the conservatism issue [3]. In recent years, ARO has demonstrated various applications, such as process scheduling [4, 5], energy systems [6], and supply chain optimization [7].

Despite the burgeoning popularity of ARO, conventional ARO models have some limitations. They fall short of leveraging uncertainty data to facilitate the decision-making processes. Moreover, their adopted uncertainty sets fail to account for the correlations or asymmetry among uncertainties. With the explosion in uncertainty data and great stride in machine learning, data-driven optimization is becoming an active research area [8-10]. Therefore, the research objective of our work is to overcome the mentioned two limitations by proposing a data-driven asymmetric uncertainty set that effectively captures correlations.

In this work, we propose a novel data-driven robust optimization framework using principal component analysis (PCA) in conjunction with kernel smoothing methods. The PCA technique is employed to acquire the principal components via eigenvalue decomposition of the covariance matrix. This way, the latent uncorrelated uncertainties behind observed uncertainty data are effectively identified. We then project uncertainty data points onto each principal component. To truthfully capture the probability distribution of latent uncertainty, nonparametric density estimation approaches are applied to the projected uncertainty data. Based on forward and backward deviation vectors [11], we further develop a data-driven polyhedron uncertainty set, in which quantile functions are used to describe confidence intervals. A notable merit of the proposed uncertainty set is that it flexibly adapts to the intrinsic structure and complexity of uncertainty data. Based upon this uncertainty set, we propose a data-driven robust optimization framework, which includes both a data-driven static robust optimization model and a data-driven ARO model. The proposed framework not only mitigates the conservatism issue of robust optimization but also enjoys considerable computational benefits. For data-driven static robust optimization, we derive a computationally tractable robust counterpart to facilitate the solution and guarantee constraint feasibility. For data-driven ARO, a decomposition-based algorithm is developed to solve the resulting multi-level optimization problem. To demonstrate the effectiveness of the proposed framework, we present three applications on optimization under uncertainty, including model predictive control (MPC), production scheduling of multipurpose batch processes, and strategic planning of process networks.

References

[1] E. N. Pistikopoulos, "Uncertainty in process design and operations," Computers & Chemical Engineering, vol. 19, pp. 553-563, 1995.

[2] 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.

[3] A. Ben-Tal, A. Goryashko, E. Guslitzer, and A. Nemirovski, "Adjustable robust solutions of uncertain linear programs," Mathematical Programming, vol. 99, pp. 351-376, 2004.

[4] Q. Zhang, M. F. Morari, I. E. Grossmann, A. Sundaramoorthy, and J. M. Pinto, "An adjustable robust optimization approach to scheduling of continuous industrial processes providing interruptible load," Computers & Chemical Engineering, vol. 86, pp. 106-119, 2016.

[5] 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.

[6] 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.

[7] 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.

[8] C. Shang, X. Huang, and F. You, "Data-driven robust optimization based on kernel learning," Computers & Chemical Engineering, vol. 106, pp. 464-479, 2017.

[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, vol. 63, pp. 3790-3817, 2017.

[10] C. Ning and F. You, "A data-driven multistage adaptive robust optimization framework for planning and scheduling under uncertainty," AIChE Journal, vol. 63, pp. 4343–4369, 2017.

[11] X. Chen, M. Sim, and P. Sun, "A robust optimization perspective on stochastic programming," Operations Research, vol. 55, pp. 1058-1071, 2007.