(126d) Data-Driven Optimization with Implicit Constraints: Application to an Ethane Steam Cracking Process

Data-driven modeling and derivative-free optimization relies on different sampling strategies that provides an initial plan for the controlled experiments on problem simulators, which is commonly known as the Design of Experiments (DoE). The goal of DoE is to provide possible candidate locations for input variables within the pre-defined box-constraints in such a way that these experiments capture a variety of system dynamics. However, the DoE is a statistical procedure and not guided by the physical information that entails an engineering process. Thus, a subset of candidate initial points generated by the DoE may result in unphysical and/or undesirable outcomes, such as an early termination of the problem simulator due to failures or solving a numerically unstable problem (stiff Ordinary Differential Equations, ODEs). This generally implies that a constraint between the optimization variables exists, in which the explicit analytical formulation of this, as a function of the input variables, is unknown to the user. As a result, the global optimization of such a system using a data-driven methodology will be hindered since the returned optimal solution may not be a feasible one.

The aim of this work is to handle implicit constraints that might exist between the input variables and guide the initial sampling strategy in such a way that the unphysical/numerically unstable combinations of input variables are filtered out, leaving only an appropriate set of samples for simulating the problem. To this end, a supervised machine learning algorithm, Support Vector Machine (SVM) is employed. The problem is formulated as nonlinear SVM classification problem where an optimal hyperplane, separating numerically stable and unstable samples, is built to implement an implicit constraint for the input space. The model is trained and cross-validated using a large set of simulated samples offline, and then incorporated in the AlgoRithms for Global Optimization of coNstrAined grey-box compUTational problems (p-ARGONAUT) [1-3] framework to assess the feasibility of the candidate points a priori to sample collection. This methodology has been tested on an ethane steam cracking process, where the continuity, energy and momentum balances are characterized by stiff ODEs in which our objective is to maximize the profitability of operation. The results show that SVM model provides a highly accurate implicit constraint and helps the grey-box optimizer to return feasible profitable solutions to the steam cracking optimization process.

References

[1] Boukouvala, F. & Floudas, C.A. ARGONAUT: AlgoRithms for Global Optimization of coNstrAined grey-box compUTational problems. Optimization Letters, 2017, 11:895-913.

[2] Boukouvala, F., Hasan, M.M.F. & Floudas, C.A. Global optimization of general constrained grey-box models: new method and its application to constrained PDEs for pressure swing adsorption. Journal of Global Optimization, 2017, 67:3-42.

[3] Beykal, B., Boukouvala, F., Floudas, C.A., Sorek, N., Zalavadia, H. & Gildin, E. Global Optimization of Grey-Box Computational Systems Using Surrogate Functions and Application to Highly Constrained Oil-Field Operations. Computers & Chemical Engineering, 2018, https://doi.org/10.1016/j.compchemeng.2018.01.005