(448b) Proto: Platform for Robust Optimization

In recent years, robust optimization has grown tremendously as a tool for incorporating parameter uncertainty into optimization models [1,2]. Robust optimization centers on the use of uncertainty sets, which define the uncertain parameter realizations which must be considered in the model. These uncertainty sets are deterministically imposed on the model, resulting in robust counterparts which replace the uncertain constraints in the original formulation. Depending on the probability distributions of the uncertain parameters and the uncertainty sets used, a wide variety of probabilistic bounds have been derived which provide guarantees on the probability of constraint violation [3-6]. These tools for handling uncertainty can be extremely valuable in answering important questions in process systems engineering [7].

In order to ease the implementation of robust optimization and the application of probabilistic bounds, a Platform for Robust OpTimizatiOn (PROTO) has been developed that performs these computations for a user based on information regarding the constraints, uncertainty sets, parameter distributions, and optimal solutions. PROTO exists as a webtool and has three main functionalities. Before solving an optimization problem, a user may provide information regarding the type of uncertainty set used and uncertain parameter distributions, if known. PROTO will then automatically determine which a priori bounds are applicable and apply them to determine the appropriate size of the uncertainty set. After solving an optimization problem, a user may provide information regarding the optimal solution and uncertain parameter distributions, and the best applicable a posteriori bound will be determined and used to calculate the probability of constraint violation. In either case, a user can also specify the bounds which they would like to utilize. Finally, if an optimization model with specified uncertain parameters is provided in GAMS format, the robust counterpart model will be formulated and a high quality robust solution will be generated at desired a posteriori probabilities using an iterative algorithm [8].

PROTO will provide probabilistic bounds for box, ellipsoidal, and polyhedral type uncertainty sets and uncertain parameters with unknown, normal, uniform, triangular, and raised cosine distributions. As some of these distributions are bounded while others are not, a natural question arose in the creation of PROTO regarding how to handle constraints with both bounded and unbounded uncertain parameters. Thus, new uncertainty sets and robust counterparts were derived for generalized uncertainty sets, which combined the interval behavior of traditional uncertainty sets for bounded parameters with unbounded uncertainty set geometries for unbounded parameters [9]. These new uncertainty sets and robust counterparts will be presented along with practical examples of the usage of PROTO which will demonstrate its potential impact for users with parameter uncertainty in their optimization problems.

[1] 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.

[2] Ben-Tal A.; El Ghaoui L.; Nemirovski A. Robust optimization. Princeton University Press, 2009.

[3] 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.

[4] 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.

[5] 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. Computers & Chemical Engineering 2017, 101, 279-311.

[6] Guzman, Y.A.; Matthews, L.R.; Floudas, C.A. New a priori and a posteriori probabilistic bounds for robust counterpart optimization: III. Exact and near-exact a posteriori expressions for known probability distributions. Computers & Chemical Engineering 2017, 103, 116-143.

[7] 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.

[8] 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.

[9] Matthews, L.R.; Guzman, Y.A.; Floudas, C.A. Generalized Robust Counterparts for Uncertain Constraints with Bounded and Unbounded Uncertain Parameters. 2017, In Preparation.