Advancements in Combinatorial and Nonlinear Optimization Methods for Process Systems Engineering

Process systems engineering aims to systematically simulate and optimize problems in chemical engineering through paradigms such as math programming and heuristics. In this thesis, we contemplate advancements in combinatorial and nonlinear optimization methods in process systems engineering, specifically addressing the following areas of research: the Resource-Constrained Project Scheduling Problem (RCPSP), deterministic global optimization (DGO), and non-linear programming (NLP). 

In resource-constrained project scheduling, project milestones, rental equipment, con?tracted labor, and product development checkpoints all conceptually impose shared due dates, which expand the concept of conventional due dates by permitting their assignment to sets of activities and by allowing multiple shared due date assignments, or none at all, to a single activity. Existing heuristic algorithms in the literature that solve the resource-constrained project scheduling problem are agnostic to shared due dates and cannot utilize information about their assignments in making scheduling decisions. To this end, we propose a novel priority rule compatible with both serial and parallel schedule generation schemes that integrates shared due date information into the heuristic scheduling decisions. We evaluate the effectiveness of our priority rule by demonstrating its dominance over existing priority rules in the literature and by comparing its performance with an exact algorithm. In comparing with the latter, we demonstrate both the computational efficiency of the heuristic as well as the quality of the solutions it produces. 

When computing bounds, spatial branch-and-bound algorithms in DGO often linearly outer approximate convex relaxations for non-convex expressions in order to capitalize on the efficiency and robustness of linear programming solvers. Considering that linear outer approximations sacrifice accuracy when approximating highly nonlinear functions and recognizing the recent advancements in the efficiency and robustness of available methods to solve optimization problems with quadratic objectives and constraints, we contemplate the construction of quadratic outer approximations of twice-differentiable convex functions for use in DGO. To this end, we present a novel cutting-plane algorithm that determines the tightest scaling parameter, α, in the second-order Taylor series approximation quadratic underestimator proposed in the literature. We use a representative set of convex functions extracted from optimization benchmark libraries to showcase– qualitatively and quantitatively–the tightness of the constructed quadratic underestimators and to demonstrate the overall computational efficiency of our algorithm. Furthermore, we extend our construction procedure to generate even tighter quadratic underestimators by allowing overestimation in infeasible polyhedral regions of optimization problems, as informed by the latter’s linear constraints. We adapt our cutting plane algorithm to address the case of non-convex difference-of-convex (d.c.) functions as well. We generalize our approach to consider a hierarchy of quadratic forms, thereby allowing the construction of even tighter underestimators. On a set of d.c. functions extracted from benchmark libraries, we demonstrate noteworthy reduction in the hypervolume between our quadratic underestimators and linear ones constructed at the same points. Additionally, we construct convex QCQP relaxations at the root node of a spatial branch-and-bound tree for a set of systematically created d.c. optimization problems in up to four dimensions, and we show that our relaxations reduce the gap between the lower bound computed by the state-of-the-art global optimization solver BARON and the optimal solution by an excess of 90%, on average. 

While the equation-oriented approach for chemical process simulation and optimization affords many advantages, one notable disadvantage is the requirement of supplying nonlinear programming (NLP) optimization algorithms with a sufficiently suitable initial point. Process modelers may expend enormous amounts of resources troubleshooting models and developing tailored initialization strategies to produce such initial points, only to discover that small changes to the model render their previously designed strategies completely ineffective. We address the problem of computing good initial points for NLP solver algorithms by utilizing distributed optimization to decompose the optimization problem into smaller problems and coordinate the solutions of the smaller problems to generate a sequence of points to be used as initial points for the NLP algorithms. We propose a framework for coupling a distributed optimization algorithm designed for non-convex NLPs with NLP algorithms such that the distributed optimization algorithm provides the initial guesses for the NLP algorithms. On a case study of an autothermal reformer process flowsheet, we demonstrate that our proposed algorithm can robustly solve all 64 (100%) instances of a parameter sweep of two model conditions, whereas default initialization strategies solve only 24 (37.5%) of the instances.