Mathematical Optimisation Advances in Process Systems Engineering

Process systems engineering is a sub-field of chemical engineering that encompasses numerous applications. It also encompasses all levels of the problem solving process, from deep conceptual understanding of the applications, through the formulation of accurate and appropriate mathematical models, to the efficient and reliable computational solution and simulation of these problems. The process systems engineer benefits from knowledge of all these areas and also an appreciation for the interaction between them. The first contribution of this thesis to the area of process system engineering is in the form of a thorough study of higher-order interior point methods. Firstly, a number of theoretical developments are presented. These developments include formal definitions of trajectories relevant to an interior point framework, the derivation of higher-order derivatives of these trajectories for anything from the linear programming case to the general convex nonlinear programming case, and examples of how these can be employed in a full algorithmic framework. Computational results of employing such frameworks highlight how these higher-order methods have a strong potential to decrease the iteration count of an interior point method. More importantly, the higher-order methods also show potential in decreasing the CPU time on certain problem types. The performance of these methods seems to positively correlate with the density of the Hessian of the objective function and of the Jacobian of the constraints, and negatively correlate with the number of general nonlinear functions, at least in the limit where this number grows very large. Theoretical results also prove that the worst-case complexity of a higher-order trajectory-following method for linear programming does not need to compromise the best known O(√n log(1 / ε)) iteration complexity of standard interior point methods. This is important in guaranteeing the performance of the new method proposed in this work when applied to increasingly large problems. Unfortunately, many practical problems are not privileged with convexity, and to deal with these, special global optimisation methods are required. An important type of nonconvex problems is the bilinear programming problem. Bilinear problems are first and foremost interesting because of their many practical applications, including pooling problems. Over a couple of chapters, this thesis contributes to the solution of nonconvex pooling problems. A core building block of many global optimisation methods is the construction and solution of convex relaxations of the nonconvex problem. Studies are made into the piecewise-linear relaxation of a reformulation of the bilinear programming problem. Additional focus is devoted to studying the migration of the solutions of these relaxations as the relaxation is squeezed through an increased number of segments in the piecewise approximation. The studies are made on a set of pooling problems from the literature. The solutions of relaxed problems do not need to be feasible with respect to the original problem. Therefore, another core building block of global optimisation methods is the identification of good feasible solutions to the original problem. Through a discretisation paradigm, three related discretisation formulations are developed for bilinear programming problems. Algorithms based on these discretisation formulations are validated by solving a number of bilinear pooling problems from the literature. Introduction of model-intrinsic binary variables into continuous optimisation problems often incurs a significant complexity increase, but as the discretisation algorithms already solve the originally nonlinear programming problems as a series of mixed-integer linear programming problems, the inclusion of such model-intrinsic binary variables is straightforward. The formulations are modified to extend beyond bilinear programming problems to be able to address general nonlinearity. This is accomplished through a combination of dimensional lifting of the original problem and relaxations of the discretisation constraints. The approach is validated on two small-scale (mixed-integer) nonlinear programming problems. The final contribution of the thesis is to the field of optimisation under uncertainty, and in particular, to the pooling problem under uncertainty. This thesis gives a thorough conceptual discussion on the role of uncertainty in the pooling problem and on the use of proxy models for solving the original application. This is in itself a major contribution because the literature appears to blindly assign and solve proxy models in place of the original problem, with no validation of the solution using the true model of the problem. The consequences of this are highlighted quantitatively by applying robust optimisation and a scenario approach to a small number of different pooling problems. The true nature of the uncertainty is assumed to be normally distributed. A model is also developed to accurately represent the true stochastic problem, and the problem is solved through a spatial branch and bound method. As such, the thesis contributes directly to all three levels of the process systems engineering hierarchy. Finally, this thesis strives to leverage the novel research ideas presented herein to unlock new opportunities and highlight promising future research directions within process systems engineering