(635b) Probabilistic Robust Optimization and a Posteriori Bounds | AIChE

# (635b) Probabilistic Robust Optimization and a Posteriori Bounds

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## Authors

Princeton University
Texas A&M University
Texas A&M University
Robust optimization is an attractive method for optimization under uncertainty in multi-scale systems engineering applications due to its deterministic handling of uncertain parameter realizations [1-8]. Key to the concept of robust optimization is the a priori simultaneous imposition of multiple possible parameter realization vectors onto uncertain constraints or an uncertain objective function [9]. Given bounded uncertainty, restricting robust optimization as simply a method for obtaining worst-case solutions [2] by imposing all possible parameter realizations onto the model will produce very conservative results. By imposing a subset of possible parameter realizations, contained in what is known as an uncertainty set [5-9], onto uncertain constraints, significantly less conservative solutions can be obtained given bounded uncertainty, and the method can be extended to unbounded uncertainty. This will yield a nonzero probability of constraint violation; the quality of a robust solution will rely heavily on the quality of applicable probabilistic expressions. In traditional one-pass robust optimization, probabilistic bounds are applied a priori in order to guide the size of uncertainty sets [10]. Fundamental theoretical advances have produced new a priori probabilistic bounds which are superior to previously existing bounds and significantly improve solutions in traditional one-pass robust optimization for models with uncertain parameters subject to unknown or known, and symmetric or asymmetric, probability distributions [11,12].

We present new a posteriori probabilistic bounds and expressions for robust optimization [11,13]. These expressions characterize robust solutions a posteriori and apply to models with uncertain parameters subject to (i) bounded but unknown probability distributions with limited information on their expected values, (ii) bounded and symmetric but otherwise unknown probability distributions, and (iii) certain known distributions. Algorithmic methods are discussed which are critical to the viability of two of the new methods. These new advances yield significantly less conservative characterizations of robust solutions when compared with existing a posteriori bounds [10,14,15]. The new a posteriori methods yield further significant improvements to traditional robust optimization techniques when applied in tandem with recent a priori bounds and within the context of iterative algorithms [16,17], as will be demonstrated with example problems.

References:

[1] Floudas, C. A., Niziolek, A. M., Onel, O., & Matthews, L. R. (2016). Multiâ?Scale Systems Engineering for Energy and the Environment: Challenges and Opportunities. AIChE Journal.

[2] Soyster, A. L. (1973). Convex programming with set-inclusive constraints and applications to inexact linear programming. Operations Research, 21(5), 1154-1157.

[3] El Ghaoui, L., & Lebret, H. (1997). Robust solutions to least-squares problems with uncertain data. SIAM Journal on Matrix Analysis and Applications, 18(4), 1035-1064.

[4] El Ghaoui, L., Oustry, F., & Lebret, H. (1998). Robust solutions to uncertain semidefinite programs. SIAM Journal on Optimization, 9(1), 33-52.

[5] Ben-Tal, A., & Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23(4), 769-805.

[6] Ben-Tal, A., & Nemirovski, A. (1999). Robust solutions of uncertain linear programs. Operations Research Letters, 25(1), 1-13.

[7] Ben-Tal, A., & Nemirovski, A. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88(3), 411-424.

[8] Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations Research, 52(1), 35-53.

[9] Li, Z., Ding, R., & Floudas, C. A. (2011). A comparative theoretical and computational study on robust counterpart optimization: I. Robust linear optimization and robust mixed integer linear optimization. Industrial & Engineering Chemistry Research, 50(18), 10567-10603.

[10] Li, Z., Tang, Q., & Floudas, C. A. (2012). A comparative theoretical and computational study on robust counterpart optimization: II. Probabilistic guarantees on constraint satisfaction. Industrial & Engineering Chemistry Research, 51(19), 6769-6788.

[11] Guzman, Y. A., Matthews, L. R., & Floudas, C. A. (2016). New a priori and a posteriori probabilistic bounds for robust counterpart optimization: I. Unknown probability distributions. Computers & Chemical Engineering, 84, 568-598.

[12] Guzman, Y. A., Matthews, L. R., & Floudas, C. A. (2016). New a priori and a posteriori probabilistic bounds for robust counterpart optimization: II. A priori bounds for known symmetric and asymmetric probability distributions. In preparation.

[13] Guzman, Y. A., Matthews, L. R., & Floudas, C. A. (2016). New a priori and a posteriori probabilistic bounds for robust counterpart optimization: III. A posteriori bounds for known probability distributions. In preparation.

[14] Paschalidis, I., Kang, S., & Li, K. (2008). Distribution-dependent robust linear optimization with asymmetric uncertainty and application to optimal control. In Proceedings of the 17th World Congress IFAC, 10069-10074.

[15] Kang, S. C., Brisimi, T. S., & Paschalidis, I. C. (2015). Distribution-dependent robust linear optimization with applications to inventory control. Annals of operations research, 231(1), 229-263.

[16] Li, Z., & Floudas, C. A. (2014). A comparative theoretical and computational study on robust counterpart optimization: III. Improving the quality of robust solutions. Industrial & engineering chemistry research,53(33), 13112-13124.

[17] Matthews, L. R., Guzman, Y. A., Onel, O., Niziolek, A. M., & Floudas, C. A. (2016). Natural gas to liquid transportation fuels: Process synthesis under feedstock and product price uncertainty using robust optimization. In preparation.