(371w) Towards Global Robust Optimisation of Non-Convex Continuous Problems: Application to Pooling Problems

Historically, flexibility analysis constitutes one of the prevailing techniques to address problems under uncertainty in the process systems engineering community. The study of robust optimisation (RO) is gaining ground in the field as an alternative approach [1]. RO is widely used to find the worst-case scenario of a problem either to guarantee constraint satisfaction under uncertainty or lacking statistical data to replace a scenario-based approach [2]. Various software has been developed to address RO problems in different optimisation environments [3,4]. Even though convex RO problems can be solved using dual reformulation, that is not always the case for non-convex problems. The growing presence of nonlinear functions in chemical engineering problems, particularly when data-driven techniques are employed, calls for the need to develop systematic techniques for the non-convex problems [5]. The past years a shift has been observed towards this direction. A local linearisation approach based on an iterative random sampling of the uncertain parameters has been proposed for different process design case studies [6]. The non-convex pooling problem under concave uncertainty is addressed in [7] following two solution paths, reformulation and cutting planes. The proposed methodology was later generalised in ROmodel [8], a Python package which allows for the modelling of robust optimisation problems. PyROS, a Pyomo solver has been developed employing a cutting-plane framework for nonlinear and non-convex models under uncertainty entailing equality constraints [9,10]. The aforementioned approaches rely on the use of state-of-the-art solvers for the global optimisation problem. Even though in the RO case the uncertainty information is integrated in the problem via the dual constraints, in the cutting plane approach the global optimisation solution is updated based on robust constraint addition that could lead to sub-optimal or even infeasible designs. To this end, the coupling of global and robust optimisation algorithms has been examined in the process systems engineering literature for the scheduling of the crude oil operations [11] and refinery planning [12]. In this work we focus on benchmark case studies of non-convex pooling problems [13] with uncertainty manifested in the inlet concentrations. The original bilinear pooling problems are reformulated using McCormick relaxations [14,15]. The examined framework is based on augmenting the deterministic spatial Branch & Bound (sBB) algorithm [16,17] with a robust infeasibility search. Finally, we evaluate the performance and quality of solutions of our proposed approach vis a vis with existing methods. [6,7,9].

Acknowledgements
Financial support under the EPSRC grants ADOPT (EP/W003317/1) and RiFtMaP (EP/V034723/1) is gratefully acknowledged by the authors.

References

[1] Zhang, Q., Grossmann, I. E., & Lima, R. M. (2016). On the relation between flexibility analysis and robust optimization for linear systems. AIChE Journal, 62(9), 3109–3123. https://doi.org/10.1002/AIC.15221

[2] Dias, L. S., & Ierapetritou, M. G. (2016). Integration of scheduling and control under uncertainties: Review and challenges. Chemical Engineering Research and Design, 116, 98–113. https://doi.org/10.1016/J.CHERD.2016.10.047

[3] Leyffer, S., Menickelly, M., Munson, T., Vanaret, C., & Wild, S. M. (2020). A survey of nonlinear robust optimization. INFOR: Information Systems and Operational Research, 58(2), 342–373. https://doi.org/10.1080/03155986.2020.1730676

[4] Zingler, A., Mitsos, A., Jungen, D., & Djelassi., H. (2023). libDIPS — Discretization-Based Semi-Infinite and Bilevel Programming Solvers – Optimization Online.https://optimization-online.org/2023/12/libdips-discretization-based-semi-infinite-and-bilevel-programming-solvers/

[5] Schweidtmann, A. M., Bongartz, D., Grothe, D., Kerkenhoff, T., Lin, X., Najman, J., & Mitsos, A. (2021). Deterministic global optimization with Gaussian processes embedded. Mathematical Programming Computation, 13(3), 553–581. https://doi.org/10.1007/S12532-021-00204-Y/TABLES/3

[6] Yuan, Y., Li, Z., & Huang, B. (2018). Nonlinear robust optimization for process design. AIChE Journal, 64(2), 481–494. https://doi.org/10.1002/AIC.15950

[7] Wiebe, J., Cecĺlio, I., & Misener, R. (2019). Robust Optimization for the Pooling Problem. Industrial and Engineering Chemistry Research, 58(28), 12712–12722. https://pubs.acs.org/doi/full/10.1021/acs.iecr.9b01772

[8] Wiebe, J., & Misener, R. (2022). ROmodel: modeling robust optimization problems in Pyomo. Optimization and Engineering, 23(4), 1873–1894. https://doi.org/10.1007/S11081-021-09703-2/TABLES/2

[9] Isenberg, N. M., Akula, P., Eslick, J. C., Bhattacharyya, D., Miller, D. C., & Gounaris, C. E. (2021). A generalized cutting-set approach for nonlinear robust optimization in process systems engineering. AIChE Journal, 67(5), e17175. https://doi.org/10.1002/AIC.17175

[10] Isenberg NM, Siirola JD, Gounaris CE Pyros (2020) A pyomo robust optimization solver for robust process design. In: 2020 Virtual AIChE Annual Meeting

[11] Li, J., Misener, R., & Floudas, C. A. (2011). Scheduling of Crude Oil Operations Under Demand Uncertainty: A Robust Optimization Framework Coupled with Global Optimization. AIChE Journal, 58, 2373–2396. https://doi.org/10.1002/aic.12772

[12] Zhang, L., Yuan, Z., & Chen, B. (2021). Refinery-wide planning operations under uncertainty via robust optimization approach coupled with global optimization. Computers & Chemical Engineering, 146, 107205. https://doi.org/10.1016/J.COMPCHEMENG.2020.107205

[13] Furini, F., Traversi, E., Belotti, P., Frangioni, A., Gleixner, A., Gould, N., Liberti, L., Lodi, A., Misener, R., Mittelmann, H., Sahinidis, N. v., Vigerske, S., & Wiegele, A. (2019). QPLIB: a library of quadratic programming instances. Mathematical Programming Computation, 11(2), 237–265. https://doi.org/10.1007/S12532-018-0147-4

[14] Misener, R., Thompson, J. P., & Floudas, C. A. (2011). APOGEE: Global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes. Computers & Chemical Engineering, 35(5), 876–892. https://doi.org/10.1016/J.COMPCHEMENG.2011.01.026

[15] Letsios, D., Baltean-Lugojan, R., Ceccon, F., Mistry, M., Wiebe, J., & Misener, R. (2020). Approximation algorithms for process systems engineering. Computers & Chemical Engineering, 132, 106599. https://doi.org/10.1016/J.COMPCHEMENG.2019.106599

[16] Tawarmalani, M., Sahinidis, N.V., 2002. Convexification and global optimization in continuous and mixed-integer nonlinear programming: Theory, Algorithms, Software, and Applications. volume 65. Springer US. https://doi.org/10.1007/978-1-4757-3532-1

[17] Boukouvala, F., Misener, R., & Floudas, C. A. (2016). Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO. European Journal of Operational Research, 252(3), 701–727. https://doi.org/10.1016/J.EJOR.2015.12.018