Comparison of MINLP formulations for global superstructure optimization

Abstract

Superstructure optimization is a powerful but computationally demanding task that can be used to select the optimal structure among many alternatives within a single optimization. In chemical engineering, such problems naturally arise in process design, where different process alternatives need to be considered simultaneously to minimize a specific objective function (e.g., production costs or global warming impact). Conventionally, superstructure optimization problems are either formulated with the Big-M or the Convex Hull reformulation approach. However, for problems containing nonconvex functions, it is not clear whether these yield the most computationally efficient formulations. We therefore compare the conventional problem formulations with less common ones (using equilibrium constraints, step functions, or multiplications of binary and continuous variables to model disjunctions) using three case studies. First, a minimalist superstructure optimization problem is used to derive conjectures about their computational performance. These conjectures are then further investigated by two more complex literature benchmarks. Our analysis shows that the less common approaches tend to result in a smaller problem size, while keeping relaxations comparably tight—despite the introduction of additional nonconvexities. For the considered case studies, we demonstrate that all reformulation approaches can further benefit from eliminating optimization variables by a reduced-space formulation. For superstructure optimization problems containing nonconvex functions, we therefore encourage to also consider problem formulations that introduce additional nonconvexities but reduce the number of optimization variables.

Molecule superstructures for computer-aided molecular and process design

“…The latter approach can be compared to a direct MINLP approach which is also used for optimizing process superstructures. 57 In our study the added complexity of the single optimization problem does not outweigh the effort of solving multiple optimization problems, especially because solving multiple problems can be parallelized efficiently.…”

A Benders decomposition framework for the optimization of disjunctive superstructures with ordered discrete decisions

“…Based on the cutting strategy and the refinement procedure shown in the previous section, a LD-BD to solve problems with external variables as complicating variables is presented in Figure 1. This algorithm offers an alternative to solve the problem stated in Equation (1) by decomposing it into an NLP subproblem that contains nonlinear constraints and optimizes continuous variables (see Equation 3), and a MIP master problem that optimizes external variables (see Equation 4).…”

“…One common aspect of many optimization problems that arise in chemical engineering is the presence of nonlinearities and discrete decisions, which often result in complex nonconvex formulations. Some examples in the field of Process System Engineering include optimal process synthesis through superstructure optimization, 1,2 optimal scheduling, 3,4 problems involving the interaction of different operational decision layers such as optimal scheduling and control, 5–9 and optimal planning, scheduling and control 10 . Due to the development of rigorous multiscale models and frameworks that integrate multiple decision layers, problems are increasing in size and complexity; thus, algorithmic and theoretical developments are still in need to alleviate the resulting computational difficulties 11 .…”

Simultaneous deterministic global flowsheet optimization and heat integration: Comparison of formulations

“…In contrast to the approach of [15], [33], this introduces additional nonlinearities, but only |P | ˆp2|C noniso | `2|H noniso | `|H iso | `|C iso |q binary and no continuous variables. In the context of superstructure optimization, we recently showed that such alternative formulations that add nonconvex terms but result in smaller problems can be beneficial if they lead to small problems and the problem is already nonconvex because of the model equations [6]. The constraints for calculating Q p in,A in this formulation are…”