(56b) Robust Optimization for Chemical Process Systems Engineering

Robust optimization (RO) is a mathematical programming-based paradigm for addressing optimization models with inherent uncertainties [1,2]. The latter may arise due to lack of precise knowledge (at the time of decision making) of model input data, decision implementation errors, or actual model structure mismatch with the real system being modeled. For chemical engineering applications, these uncertainties naturally map to the use of empirical data for kinetics and heat/mass transfer, imperfect control via actuators, or unknown material phases inside the system, among other sources. There has been less focus in recent years on the application of RO methodologies to chemical process systems engineering (PSE). This is largely due to the fact that PSE models often include complex nonlinear and non-convex constraints, which originate from many physical and chemical state equations. This means that reformulation-based solution methods for solving RO models may lead to either overly conservative or even robust infeasible solutions. Recently, there has been increased interest in the application of RO to nonlinear and non-convex systems [3,4]. Similarly, there have been novel methods and applications of the RO paradigm to nonlinear process systems engineering models, including a general nonlinear programming robust counterpart formulation [5], a robust counterpart formulation with local linearization of nonlinear uncertain constraints paired with a novel sampling algorithm [6], and the application of a robust cutting-plane algorithm to the pooling problem [7].

To date, there is no holistic approach in the RO literature for identifying robust solutions in nonlinear, non-convex, PSE-type models that insures feasibility against the entire uncertainty space and handles the general recourse ability afforded to PSE systems via process control schemes. The derivation of robust counterparts, e.g., via duality-based reformulations to single-level optimization problems, is a prohibitively complex task for the average process modeler, and the resulting models may not be trivial to solve, even with modern NLP solvers. For this reason, we propose an alternative algorithmic approach for identifying robust solutions to uncertain PSE models. Our approach is based on a robust cutting-set algorithm [8], which we generalized to apply to models with non-convexities in both the decision variables and uncertain parameters, including two-stage models featuring first- and second-stage (e.g., design and control) variables via general non-linear decision rule relationships. We have codified this algorithm in PyROS, a Python-based, open-source tool that allows owners of deterministic models with characterized uncertainties to easily identify robust solutions under various selected uncertainty set geometries [9]. In this talk, we demonstrate the use of PyROS on addressing increasingly complex case studies from the PSE domain, including reactor-separator, reactor-heater, and solvent-based CO₂ capture flowsheets [10].

References

[1] Dimitris Bertsimas, David B. Brown, and Constantine Caramanis. Theory and applications of robust optimization. SIAM review, 53(3), 464-501, 2011.

[2] Aharon Ben-Tal, Laurent El Ghaoui, and Arkadi Nemirovski. Robust optimization. Vol. 28. Princeton University Press, 2009.

[3] Dimitris Bertsimas, Omid Nohadani, and Kwong Meng Teo. Nonconvex robust optimization for problems with constraints. INFORMS Journal on Computing, 22(1):44-58, 2010.

[4] Sven Leyffer, Matt Menickelly, Todd Munson, Charlie Vanaret & Stefan M. Wild. A survey of nonlinear robust optimization, INFOR: Information Systems and Operational Research, 58:2, 342-373, 2020.

[5] Yin Zhang. General robust-optimization formulation for nonlinear programming. Journal of Optimization Theory and Applications, 132(1):111-124, 2007.

[6] Yuan Yuan, Zukui Li, and Biao Huang. Nonlinear robust optimization for process design. AIChE Journal, 64(2):481-494, 2018.

[7] Johannes Wiebe, Ines Cecilio, and Ruth Misener. Robust optimization for the pooling problem. Industrial & Engineering Chemistry Research, 2019.

[8] Almir Mutapcic and Stephen Boyd. Cutting-set methods for robust convex optimization with pessimizing oracles. Optimization Methods & Software, 24(3):381-406, 2009.

[9] Natalie M. Isenberg, John D. Siirola, Chrysanthos E. Gounaris, PyROS: A Pyomo Robust Optimization Solver for Robust Process Design. In preparation.

[10] Natalie M. Isenberg, Paul Akula, John C. Eslick, Debangsu Bhattacharyya, David C. Miller, and Chrysanthos E. Gounaris. A generalized cutting-set approach for nonlinear robust optimization in process systems engineering applications. Under Review.