Mixed-Integer Nonlinear Programming Algorithms Based on Discrete Convex Analysis with Applications to Integrated Decision-Making Problems

Abstract

This PhD thesis presents the development and application of new optimization models and algorithms for the solution of relevant Mixed Integer Nonlinear Programming (MINLP) and Generalized Disjunctive Programming (GDP) optimization problems arising in the fields of process design, simultaneous design and control, and simultaneous scheduling and control. The algorithms proposed throughout this thesis rely on a new optimization paradigm: incorporating concepts from discrete convex analysis (DCA) into the development of efficient MINLP and GDP optimization algorithms, rather than relying solely on conventional or general-purpose MINLP or GDP solvers. The main feature of DCA-based algorithms is that they may trigger a more efficient exploration of the integer and some binary or Boolean variables in the formulation known as ordered discrete decisions, leading to alternative local solutions that may not be attained with traditional local deterministic techniques. The advantages of these algorithms are demonstrated through the solution of a variety of optimization problems relevant in chemical engineering. The first problem considered in this thesis is the rate-based optimal design of catalytic distillation columns. This thesis shows the application of a DCA-based algorithm to address this problem, known as the Discrete-Steepest Descent Algorithm (D-SDA). To the author’s knowledge, this is an optimization problem that has not been attempted to be solved using deterministic MINLP optimization strategies in the past. This thesis also explores the application of the D-SDA as a logic-based solver (LD-SDA) to address GDP problems and introduces a new DCA-based technique called logic-based Discrete-Benders Decomposition (LD-BD). This method was designed following logic-based Benders Decomposition (LBBD) principles, which aim to extend the Benders Decomposition (BD) idea to a broader class of optimization problems, by giving the user freedom on the design of Benders cuts tailored for each application. To the author’s knowledge, LD-BD is the first algorithm that combines DCA theory with LBBD principles. Another variant of the D-SDA proposed in this work is the Discrete-Steepest Descent Algorithm with Variable Bounding (DSDA-VB), which is used as the core method within a hybrid stochastic (i.e., metaheuristic)-deterministic algorithm for optimal design of process flowsheets. Unlike previous studies that propose hybrid deterministic and stochastic algorithms in sequential and nested arrangements, this thesis proposes a parallel configuration to perform the hybridization. This thesis also brings novelty by extending the application of DCA-based optimization tools to the Mixed-Integer Dynamic Optimization (MIDO) domain. The first MIDO problem considered in this work is the optimal design and operation of CD units considering discrete and continuous design and operation variables combined with rigorous non-linear dynamic process models. The problem is solved with an enhanced modular D-SDA framework. The key novelty in this study is that optimal process design and dynamic transitions between different product grades in CD units are simultaneously optimized using a deterministic optimization framework. Another MIDO problem considered in this thesis arises in the field of simultaneous scheduling and control. This work proposes a general discrete-time simultaneous scheduling and dynamic optimization (SSDO) formulation based on the State Task Network (STN) representation. This formulation explicitly considers variable processing times, which is a key aspect in the integration of scheduling and control decisions. The resulting problem is solved using LD-SDA. This thesis also presents the first application of a LBBD technique in the field of SSDO applied to STN processes with a discrete-time scheduling formulation, through the herein proposed Multicut LD-BD (MLD-BD) algorithm. MLD-BD builds upon LD-BD and improves its performance by incorporating multiple cuts per iteration, a pruning strategy, and a cut-off technique. Different case studies show that the herein proposed algorithms explore the feasible region of ordered discrete decisions more efficiently than general purpose MINLP solvers, leading to more profitable solutions in shorter computational times. Overall, this thesis brings advances to the field of MINLP optimization by 1) introducing and demonstrating the advantages of new DCA-based algorithms when solving challenging problems in the fields of optimal design, and 2) extending these algorithms to the field of MIDO optimization, with optimal process integration applications.