(19f) A Sequential Approach to Global Flowsheet Optimization Using Mccormick Relaxations

Local optimization methods are widely employed for flowsheet optimization, using formulations based on either the equation-oriented mode or the sequential modular mode of flowsheet simulation, or some intermediate version [5]. While in the former case, all model variables are considered as optimization variables and all model equations as equality constraints, the latter approaches reduce the problem size by operating on a reduced set of variables that typically correspond to the degrees of freedom in designing the process, augmented by the variables associated with tear streams (in case of infeasible path methods), while the majority of the model equations are `hidden' from the optimizer in external modules [1]. Deterministic global methods for flowsheet optimization have so far almost exclusively relied on an equation-oriented approach (cf., e.g., [11]), because the branch-and-bound-based (B&B) solvers used require explicit access to the model equations in order to supply convex relaxations [7, 10]. As the flowsheet size and model complexity increases, this approach can get challenging because of the exponential worst-case runtime of B&B solvers, and because the user generally has to supply sufficiently tight bounds on a large number of variables. From this perspective, approaches that operate in a reduced space seem particularly attractive for deterministic global optimization.

In this work, we consider a method for global optimization of flowsheets that is similar to the sequential modular infeasible path method employed in local optimization in the sense that the majority of the model variables and equations are moved to external modules, while only few are left to the optimizer. While a similar approach was originally introduced using interval methods [3], a recent modification based on McCormick relaxations has been shown to enable significant reductions in computational time compared to equation-oriented formulations [2]. This approach makes use of the automatic propagation of McCormick relaxations [6] and their subgradients through computer codes [8] that allow to supply tight relaxations for functions that are not explicitly handled by the optimizer, but rather act as a black box from which function values or relaxations and their subgradients can be queried as needed. Through an appropriate selection of the equations to be left to the optimizer, the applicability of the relaxation propagation is ensured. A possible alternative would be to optimize only in the degrees of freedom, which would require the relaxation of implicit functions [9]. A simple B&B solver has been implemented that can treat external module definitions. The modules are implemented as template functions, and the propagation of relaxations is conducted using MC++ [4]. The application of this reduced-space formulation is demonstrated for simple process flowsheets. We compare solution times of the reduced-space formulation with those of the conventional equation-oriented formulation and discuss the influence of the selection of variables and constraints left to the optimizer.

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