Optimization-based Methodologies Research Articles

A practical optimization framework for political redistricting: A case study in Arizona

Following the decennial census, each state in the U.S. redraws its congressional and state legislative district boundaries, which must satisfy various legal criteria. For example, Arizona’s Constitution describes six legal criteria, including contiguity, population balance, competitiveness, compactness, and the preservation of communities of interest, political subdivisions and majority–minority districts, each of which is to be enforced “to the extent practicable”. Optimization algorithms are well suited to draw district maps, although existing models and methods have limitations that inhibit their ability to draw legally-valid maps. Adapting existing optimization methods presents two major challenges: the complexity of modeling to achieve multiple and subjective criteria, and the computational intractability when dealing with large redistricting input graphs. In this paper, we present a multi-stage optimization framework tailored to redistricting in Arizona. This framework combines key features from existing methods, such as a multilevel algorithm that reduces graph input sizes and a larger local search neighborhood that encourages faster exploration of the solution space. This framework heuristically optimizes geographical compactness and political competitiveness while ensuring that other criteria in Arizona’s Constitution are satisfied relative to existing norms. Compared to Arizona’s enacted map (CD118) to be used until 2032, the most compact map produced by the algorithm is 41% more compact, and the most competitive map has five more competitive districts. To enable accessibility and to promote future research, we have created Optimap, a publicly accessible tool to interact with a part of this framework. Beyond the creation of these maps, this case study demonstrates the positive impact of adapting optimization-based methodologies for political redistricting in practice.

Simulation-Driven and Optimization-Based Design of an Architectural Building: A Case Study of a Space Tourism Building in the UAE

In the constantly changing field of architectural design and meeting the evolving demands of the space tourism industry, this study presents a case study focused on the design of a space tourism building in Sharjah, United Arab Emirates (UAE). The main objective of this study is to showcase the integration of simulation-driven and optimization-based methodologies in architectural design to anticipate forthcoming challenges and resolve design-related issues or constraints. This study employs a range of computational tools to optimize various design parameters, including sun path, radiation, shadow, outdoor comfort, and wind simulation, to conduct a more thorough assessment in the process of choosing the best-optimized design. In addition, a finite element analysis (FEA) was carried out to gain insights into the structural behavior of the chosen design under diverse physical conditions. This integration marks a paradigm shift in how architectural projects are conceptualized, developed, and realized by addressing complex challenges and enhancing the functionality, sustainability, and performance of architectural buildings. The findings of this case study demonstrate and prove the potential of simulation-driven and optimization-based design approaches in determining the future of architectural designs. As the UAE strives to become a center for space-related activities, this research helps to drive the realization of cutting-edge, sustainable, and user-centric architectural solutions, paving the way for the next generation of space tourism facilities.

Source Characterization of Multiple Reactive Species at an Abandoned Mine Site Using a Groundwater Numerical Simulation Model and Optimization Models.

The most important first step in the management and remediation of contaminated groundwater aquifers is unknown contaminant source characterization. Often, the hydrogeological field data available for accurate source characterization are very sparse. In addition, hydrogeological and geochemical parameter estimates and field measurements are uncertain. Particularly in complex contaminated sites such as abandoned mine sites, the geochemical processes are very complex and identifying the sources of contamination in terms of location, magnitude, and duration, and determination of the pathways of pollution become very difficult. The reactive nature of the contaminant species makes the geochemical transport process very difficult to model and predict. Additionally, the source identification inverse problem is often non-unique and ill posed. This study is about developing and demonstrating a source characterization methodology for a complex contaminated aquifer with multiple reactive species. This study presents linked simulation optimization-based methodologies for characterization of unknown groundwater pollution source characteristics, i.e., location, magnitude and duration or timing. Optimization models are solved using an adaptive simulated annealing (ASA) optimization algorithm. The performance of the developed methodology is evaluated for different complex scenarios of groundwater pollution such as distributed mine waste dumps with reactive chemical species. The method is also applied to a real-life contaminated aquifer to demonstrate the potential applicability and optimal characterization results. The illustrative example site is a mine site in Northern Australia that is no longer active.

Optimization-based planning of local energy systems -bridging the research-practice gap

Optimization-based planning of local energy systems – though increasingly mature from a methodological perspective – is not commonly applied in practice. This paper synthesizes learnings from 4 case studies – focused on 4 different sites – in which optimization-based methodologies have been applied to the planning of local energy systems. The aim is to generate insights to facilitate the more effective application of optimization-based methodologies to local energy systems planning in practice. The results indicate that an intensively iterative methodology is critical not only as a basis for adapting the analysis based on stakeholder input, but also to facilitate learning on the part of stakeholders with regard to the value and limitations of the approach and the results. With regard to optimization methodologies, in particular temporal decomposition methodologies are identified as critical to preserving computational tractability in the optimization of complex technical systems, especially those featuring networks and those with many energy carriers or technology options. It is suggested that the methodological tailoring of an optimization model to a specific case and the calculation/visualization of key indicators can be largely automated, which could significantly accelerate future studies and reduce the knowledge required for their execution.

Synthesis of Heat-Integrated Water Allocation Networks: A Meta-Analysis of Solution Strategies and Network Features

Industries consume large quantities of energy and water in their processes which are often considered to be peripheral to the process operation. Energy is used to heat or cool water for process use; additionally, water is frequently used in production support or utility networks as steam or cooling water. This enunciates the interconnectedness of water and energy and illustrates the necessity of their simultaneous treatment to improve energy and resource efficiency in industrial processes. Since the seminal work of Savulescu and Smith in 1998 introducing a graphical approach, many authors have contributed to this field by proposing graphically- or optimization-based methodologies. The latter encourages development of mathematical superstructures encompassing all possible interconnections. While a large body of research has focused on improving the superstructure development, solution strategies to tackle such optimization problems have also received significant attention. The goal of the current article is to study the proposed methodologies with special focus on mathematical approaches, their key features and solution strategies. Following the convention of Jeżowski, solution strategies are categorized into: decomposition, sequential, simultaneous, meta-heuristics and a more novel strategy of relaxation/transformation. A detailed, feature-based review of all the main contributions has also been provided in two tables. Several gaps have been highlighted as future research directions.

Variational inequality approach to enforcing the non-negative constraint for advection–diffusion equations

Predictive simulations are crucial for the success of many subsurface applications, and it is highly desirable to obtain accurate non-negative solutions for transport equations in these numerical simulations. To this end, optimization-based methodologies based on quadratic programming (QP) have been shown to be a viable approach to ensuring discrete maximum principles and the non-negative constraint for anisotropic diffusion equations. In this paper, we propose a computational framework based on the variational inequality (VI) which can also be used to enforce important mathematical properties (e.g., maximum principles) and physical constraints (e.g., the non-negative constraint). We demonstrate that this framework is not only applicable to diffusion equations but also to non-symmetric advection–diffusion equations. An attractive feature of the proposed framework is that it works with any weak formulation for the advection–diffusion equations, including single-field formulations, which are computationally attractive. A particular emphasis is placed on the parallel and algorithmic performance of the VI approach across large-scale and heterogeneous problems. It is also shown that QP and VI are equivalent under certain conditions. State-of-the-art QP and VI solvers available from the PETSc library are used on a variety of steady-state 2D and 3D benchmarks, and a comparative study on the scalability between the QP and VI solvers is presented. We then extend the proposed framework to transient problems by simulating the miscible displacement of fluids in a heterogeneous porous medium and illustrate the importance of enforcing maximum principles for these types of coupled problems. Our numerical experiments indicate that VIs are indeed a viable approach for enforcing the maximum principles and the non-negative constraint in a large-scale computing environment. Also provided are Firedrake project files as well as a discussion on the computer implementation to help facilitate readers in understanding the proposed framework.

Large-Scale Optimization-Based Non-negative Computational Framework for Diffusion Equations: Parallel Implementation and Performance Studies

It is well-known that the standard Galerkin formulation, which is often the formulation of choice under the finite element method for solving self-adjoint diffusion equations, does not meet maximum principles and the non-negative constraint for anisotropic diffusion equations. Recently, optimization-based methodologies that satisfy maximum principles and the non-negative constraint for steady-state and transient diffusion-type equations have been proposed. To date, these methodologies have been tested only on small-scale academic problems. The purpose of this paper is to systematically study the performance of the non-negative methodology in the context of high performance computing (HPC). PETSc and TAO libraries are, respectively, used for the parallel environment and optimization solvers. For large-scale problems, it is important for computational scientists to understand the computational performance of current algorithms available in these scientific libraries. The numerical experiments are conducted on the state-of-the-art HPC systems, and a single-core performance model is used to better characterize the efficiency of the solvers. Our studies indicate that the proposed non-negative computational framework for diffusion-type equations exhibits excellent strong scaling for real-world large-scale problems.

Optimal process and control design under uncertainty: A methodology with robust feasibility and stability analyses

This paper presents a new methodology for the optimal process and control design of dynamic systems under uncertainty. Robust feasibility and stability analyses are incorporated within the proposed methodology to ensure process dynamic operability and asymptotic stability. These analyses are formulated as convex mathematical problems; thus, the present approach is computationally attractive since it does not require the solution of an MINLP to evaluate dynamic feasibility and stability as it has been proposed by recent dynamic optimization-based methodologies. A norm-bounded metric based on Structured Singular Value (SSV) analysis is employed to estimate the worst-case deviation in the process constraints in the presence of critical realizations in the disturbances. The robust stability test is based on Lyapunov theory and guarantees process asymptotic stability. Accordingly, the optimal process and control design alternative obtained by the method proposed here is dynamically feasible and asymptotically stable since it accommodates the most critical realizations in the disturbances. A ternary distillation system featuring a rigorous tray-by tray process model is used to illustrate the application of the proposed method.

A dynamic mathematical model for monoclonal antibody N‐linked glycosylation and nucleotide sugar donor transport within a maturing Golgi apparatus

Monoclonal antibodies (mAbs) are one of the most important products of the biopharmaceutical industry. Their therapeutic efficacy depends on the post-translational process of glycosylation, which is influenced by manufacturing process conditions. Herein, we present a dynamic mathematical model for mAb glycosylation that considers cisternal maturation by approximating the Golgi apparatus to a plug flow reactor and by including recycling of Golgi-resident proteins (glycosylation enzymes and transport proteins [TPs]). The glycosylation reaction rate expressions were derived based on the reported kinetic mechanisms for each enzyme, and transport of nucleotide sugar donors [NSDs] from the cytosol to the Golgi lumen was modeled to serve as a link between glycosylation and cellular metabolism. Optimization-based methodologies were developed for estimating unknown enzyme and TP concentration profile parameters. The resulting model is capable of reproducing glycosylation profiles of commercial mAbs. It can further reproduce the effect gene silencing of the FucT glycosylation enzyme and cytosolic NSD depletion have on the mAb oligosaccharide profile. All novel elements of our model are based on biological evidence and generate more accurate results than previous reports. We therefore believe that the improvements contribute to a more detailed representation of the N-linked glycosylation process. The overall results show the potential of our model toward evaluating cell engineering strategies that yield desired glycosylation profiles. Additionally, when coupled to cellular metabolism, this model could be used to assess the effect of process conditions on glycosylation and aid in the design, control, and optimization of biopharmaceutical manufacturing processes.