(635c) A Comparative Analysis of Robust Optimization and Scenario-Based Approaches for Optimization Under Uncertainty

It is common in mathematical optimization problems to assume that parameters will retain constant values, when in reality, these parameters may have potential to vary greatly. This uncertainty can have a tremendous impact on the objective function of the optimal solution, or on overall model feasibility. As noted in a recent perspective article on multi-scale engineering for energy and the environment, the inclusion of uncertainty in process systems engineering problems will be crucial in addressing major challenges ahead [1]. To address these issues, different approaches for optimization under uncertainty have been proposed for a range of applications [2,3]. Two widely used approaches include robust optimization [4,5] and stochastic programming using scenarios. The former has widely been considered overly conservative, as the optimal solution must remain feasible for an uncertainty set of parameter values. Scenario-based approaches, on the other hand, generate scenarios that include a subset of possible parameter values, and find the expected objective value over that set of scenarios, albeit with large computational times and limited information on the probability of constraint violation.

Through multiple applications in process systems engineering, the two approaches for handling uncertain parameters are compared for optimization models with uncertainty in the objective function. For problems with uncertainty in the objective function, the probability of constraint violation can be defined for the scenario-based approach in order to compare the solutions from each approach at various levels of conservatism. In order to cover the entire uncertain space with probabilistic guarantees, scenarios will be generated based on discretizations between the bounds of the uncertain parameters. Multiple discretization levels will be assumed in order to measure the impact of discretization on the performance of a scenario-based approach. Robust solutions will also be generated using a priori and a posteriori probabilistic bounds and iterative methods for improving the quality of robust solutions [6].

Due to recent advances in probabilistic bounds for robust optimization [7-9], it will be shown that robust optimization is a competitive, or even superior, approach for optimization under uncertainty. Optimal solutions and computational time will be compared for the two approaches using example problems and will be discussed in the context of large-scale optimization problems using robust results. These results will demonstrate that robust optimization provides robust solutions in a fraction of the computational time required for the scenario-based approach while giving as good or better objective function values at the same probabilities of constraint violation, based on the assumptions made regarding the uncertain parameter distributions.

[1] Floudas, C. A.; Niziolek, A. M.; Onel, O.; Matthews, L. R. Multi-scale systems engineering for energy and the environment: Challenges and opportunities. AIChE Journal 2016, 62 (3), 602-623.

[2] Verderame, P. M.; Elia, J. A.; Li, J.; Floudas, C. A. Planning and scheduling under uncertainty: A review across multiple sectors. Industrial & Engineering Chemistry Research 2010, 49, 3993-4017.

[3] Grossmann, I. E.; Apap, R. M.; Calfa, B. A.; Garcia-Herreros, P.; Zhang, Q. Recent advances in mathematical programming techniques for the optimization of process systems under uncertainty. Computers and Chemical Engineering. (2016). In Press.

[4] Li, Z.; Ding, R.; Floudas, C. A. A Comparative Theoretical and Computational Study on Robust Counterpart Optimization: I. Robust Linear Optimization and Robust Mixed Integer Linear Optimization. Industrial & Engineering Chemistry Research 2011, 50, 10567-10603.

[5] Li, Z.; Tang, Q.; Floudas, C. A. A Comparative Theoretical and Computational Study on Robust Counterpart Optimization: II. Probabilistic Guarantees on Constraint Satisfaction. Industrial & Engineering Chemistry Research 2012, 51 (19), 6769-6788.

[6] Li, Z.; Floudas, C. A. A Comparative Theoretical and Computational Study on Robust Counterpart Optimization: III. Improving the Quality of Robust Solutions. Industrial & Engineering Chemistry Research 2014, 53 (33), 13112-13124.

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

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

[9] Guzman, Y. A; Matthews, L. R; Floudas, C. A New a priori and a posteriori probabilistic bounds for robust counterpart optimization: III. A posteriori bounds for known probability distributions. 2016, In Preparation.