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

Abstract

This study introduces the logic‐based discrete‐Benders decomposition (LD‐BD) for Generalized Disjunctive Programming (GDP) superstructure problems with ordered Boolean variables. The key idea is to obtain Benders cuts that use neighborhood information of a reformulated version of Boolean variables. These Benders cuts are iteratively refined, which guarantees convergence to a local optimum. A mathematical case study, the optimization of a network with Continuous Stirred‐Tank Reactors (CSTRs) in series, and a large‐scale problem involving the design of a distillation column are considered to demonstrate the features of LD‐BD. The results from these case studies have shown that the LD‐BD method exhibited good performance by finding attractive locally optimal solutions relative to existing logic‐based solvers for GDP problems. Based on these tests, the LD‐BD method is a promising strategy to solve optimal synthesis problems with ordered discrete decisions emerging in chemical engineering applications.

Simultaneous design and NMPC control under uncertainty and structural decisions: A discrete‐steepest descent algorithm

“…For future work, we aim to extend the D‐SDA framework for the integration of design and NMPC‐based control for more complex systems, for example, reactive distillation columns. An improved search strategy based on a GBD 38 will be incorporated into the present framework to avoid the use of auxiliary function in the algorithm (e.g., ) and to reduce the computational cost.…”

“…For future work, we aim to extend the D-SDA framework for the integration of design and NMPC-based control for more complex systems, for example, reactive distillation columns. An improved search strategy based on a GBD38 will be incorporated into the present framework to avoid the use of auxiliary function in the algorithm (e.g., Ψ P ) and to reduce the computational cost.AUTHOR CONTRIBUTIONSOscar Palma-Flores: conceptualization (equal); formal analysis (lead); investigation (equal); methodology (equal); validation (lead); visualization (lead); writingoriginal draft (lead). Luis A. Ricardez-Sandoval: conceptualization (lead); formal analysis (lead); funding acquisition (equal); investigation (lead); methodology (lead); project administration (equal); supervision (lead); validation (equal); writingreview and editing (lead).…”

Discrete-Time Network Scheduling and Dynamic Optimization of Batch Processes with Variable Processing Times through Discrete-Steepest Descent Optimization

“…The ordered discrete decisions may not appear explicitly in the formulation; thus, a reformulation of the problem is usually required . In general, ordered discrete decisions may appear as explicit integer variables or as groups of binary/Boolean variables Y i defined over ordered sets (i.e., i ∈ Ordered set) plus an assignment constraint (i.e., exactly k Y i are 1/True) . In this case, there is a mixture of both: batching variables n i,j , ∀( i , j ) ∈ IJ × (eq ) are explicitly ordered discrete decisions, whereas Boolean variables Y τ i,j ,i,j , ∀( i , j ) ∈ IJ × are ordered discrete decisions with respect to index τ i,j ∈ Τ i,j (eq ), where Τ i,j can be regarded as an ordered set.…”

“…These special variables have only been studied in the context of optimal process design and simultaneous design and control. 27,28 It has been shown that general-purpose MINLP solvers are unable to effectively capture these ordered structures in their optimization procedures, leading to suboptimal solutions and lengthy computational times. To overcome this limitation, the Discrete-Steepest Descent Algorithm (D-SDA) was developed as a specialized decomposition strategy to effectively optimize ordered discrete decisions in MINLP superstructure design problems.…”

“…In the case of discrete-time network SSDO, examples of ordered discrete decisions include batching-assignment variables (or batching variables for brevity), which decide the number of times a given task is executed in a given unit, or the number of time points involved in the execution of a task to represent its duration. These special variables have only been studied in the context of optimal process design and simultaneous design and control. , It has been shown that general-purpose MINLP solvers are unable to effectively capture these ordered structures in their optimization procedures, leading to suboptimal solutions and lengthy computational times. To overcome this limitation, the Discrete-Steepest Descent Algorithm (D-SDA) was developed as a specialized decomposition strategy to effectively optimize ordered discrete decisions in MINLP superstructure design problems. , From a theoretical perspective, the main difference between the D-SDA and general-purpose MINLP/GDP solvers is that D-SDA does not rely on traditional optimality and convexity requirements for ordered discrete decisions, e.g., a MINLP is “convex” if the relaxed MINLP is convex.…”