Individual Subproblems Research Articles

GENERAL PRINCIPLES OF FORMALIZATION OF TECHNOLOGICAL PROCESS CONTROL OF MINING PRODUCTION IN A DYNAMIC DISTRIBUTED SYSTEM

Context. The problem of synthesis, modeling, and analysis of automated control of complex technological processes of mining production as a dynamic structure with distributed parameters. Objective. On the example of the technological line of ore beneficiation, the general principles of formalization of control of mining production processes as a dynamic system with distributed parameters are considered. Method. The modeling of interactions between individual components of the control system is carried out using the methods of coordinated distributed control. In accordance with this approach, the technological line is decomposed into a set of separate subsystems (technological units, enrichment cycles). Under these circumstances, the solution to the global optimization problem is also decomposed into a corresponding set of individual subproblems of optimizing the control of subsystems. To solve the global problem, this formulation uses a two-level structure with coordinating variables that are fed to the input of local control systems for technological units and cycles. At the lower level of control, sets of subtasks have independent solutions, coordinated by the coordinating variables formed at the upper level. Results. The paper proposes a method for forming control of a distributed system of technological units of an ore dressing line based on the decomposition of the dynamics of the distributed system into time and space components. In the spatial domain, the control synthesis problem is solved as a sequence of approximation problems of a set of spatial components of the dynamics of the controlled system. In the time domain, the solution of the control synthesis problem is based on the methods of synthesizing control systems with concentrated parameters. Conclusions. The use of the proposed approach to the formation of technological process management at mining enterprises of the Kryvyi Rih iron ore basin will improve the quality of iron ore concentrate supplied to metallurgical processing, increase the productivity of technological units and reduce energy consumption.

A reinforcement learning-driven adaptive decomposition algorithm for multi-objective hybrid seru system scheduling considering worker transfer

The application of hybrid seru manufacturing in real-world production often necessitates the implementation of a worker transfer strategy to overcome limitations in skill levels and realize the co-optimization of production efficiency and worker workload. Therefore, this study focuses on hybrid seru system scheduling considering worker transfer (HSSWT) to minimize both the makespan and the total labor time. The presence of multiple optimization objectives and coupled subproblems results in a considerably large search space, which poses challenges that existing optimization frameworks cannot effectively address. Thus, we propose a reinforcement learning-driven adaptive decomposition algorithm (RLADA) to address this issue. First, to minimize the search space, the HSSWT is decomposed into two subproblems, each utilizing distinct search strategies to explore specific areas. Elite-driven population evolution and a lower bound-guided greedy search are proposed to solve the individual subproblems. Second, a deep Q-network has been developed to adaptively select the subproblem to optimize in each generation by dynamically dividing the objective space. Finally, to narrow the search space, the lower bound is used to filter out unpromising solutions, thereby avoiding invalid searches. Additionally, the lower bound is utilized to determine the search depth according to the optimization margin, helping to conserve computing resources. Extensive comparative results demonstrate the effectiveness of the specialized designs, and the RLADA outperforms state-of-the-art algorithms in terms of solution convergence and diversity when addressing the HSSWT.

Understanding collaboration in sub-structured teams through a computational model of set-based concurrent engineering

Coordinating the collaboration within sub-structured teams created for large engineering projects presents several challenges. Set-based concurrent engineering (SBCE), by maintaining several candidate designs for each subproblem throughout the process, may alter the best ways to coordinate subteams in these projects, compared to point-based concurrent engineering (PBCE) processes in which only one candidate design is maintained for each subproblem. Previous studies have introduced SBCE to industry with positive results and have corroborated its stated benefits through computational study, but researchers have yet to systematically study how SBCE interacts with sub-structured teams. This work augments a multi-agent model developed to simulate and study SBCE, the Point/Set-Organized Research Teams (PSORT) platform, to model sub-structured teams with different forms of intra-subteam collaboration, including joint iteration, design sharing, or no collaboration at all. Design projects are then simulated using these collaboration methods with PBCE and SBCE conditions to investigate how SBCE affects team performance under different modes of intra-subteam collaboration. These studies find that SBCE enables nominal subteams to combine the efforts of multiple designers on individual subproblems with minimal losses, while high levels of collaboration during development are found to assist designers in overcoming design barriers, which SBCE facilitates by maintaining many design options.

A distributed randomized method for the identification of switched ARX systems

SummaryThe identification of switched systems amounts to a mixed integer nonlinear optimization problem, where the continuous variables are associated to the model parameterizations of the different modes, and the discrete ones are related to the switching signal (each data sample is assigned to a mode, and switching occurs when the mode assignment changes over time). In the batch form of the identification problem, the combinatorial complexity increases exponentially with the size of the training set, which makes the precise identification of the switching signal the most challenging task in the identification problem. To tackle this complexity we propose a distributed optimization approach, based on the solution of multiple instances of a much simpler problem, where switching can occur only at specific time instants (different for each subproblem), and an information sharing mechanism that preserves likely switching times to improve the local solutions. We employ an adapted version of a previously developed randomized algorithm to solve the individual subproblems. Another important feature of the proposed method is an a posteriori heuristic correction method, that is applied to further refine the switching locations based on the estimated local models before the information sharing phase. The performance of the proposed algorithm is analyzed and compared with other methods on synthetic datasets.

Efficient numerical solution of the Fokker–Planck equation using physics-conforming finite element methods

Abstract We consider the Fokker–Planck equation (FPE) for the orientation probability density of fiber suspensions. Using the continuous Galerkin method, we express the numerical solution in terms of Lagrange basis functions that are associated with N nodes of a computational mesh for a domain in the 3D physical space and M nodes of a mesh for the surface of a unit sphere representing the configuration space. The NM time-dependent unknowns of our finite element approximations are probabilities corresponding to discrete space locations and orientation angles. The framework of alternating-direction methods enables us to update the numerical solution in parallel by solving N evolution equations on the sphere and M three-dimensional advection equations in each (pseudo-)time step. To ensure positivity preservation as well as the normalization property of the probability density, we perform algebraic flux correction for each equation and synchronize the correction factors corresponding to different orientation angles. The velocity field for the spatial advection step is obtained using a Schur complement method to solve a generalized system of the incompressible Navier–Stokes equations (NSE). Fiber-induced subgrid-scale effects are taken into account using an effective stress tensor that depends on the second- and fourth-order moments of the orientation density function. Numerical studies are performed for individual subproblems and for the coupled FPE-NSE system.

Interface flux recovery framework for constructing partitioned heterogeneous time‐integration methods

AbstractA common approach for the development of partitioned schemes employing different time integrators on different subdomains is to lag the coupling terms in time. This can lead to accuracy issues, especially in multistage methods. In this article, we present a novel framework for partitioned heterogeneous time‐integration methods, which allows the coupling of arbitrary multistage and multistep methods without reducing their order of accuracy. At the core of our approach are accurate estimates of the interface flux obtained from the Schur complement of an auxiliary monolithic system. We use these estimates to construct a polynomial‐in‐time approximation of the interface flux over the current time coupling window. This approximation provides the interface boundary conditions necessary to decouple the subdomain problems at any point within the coupling window. In so doing our framework enables a flexible choice of time‐integrators for the individual subproblems without compromising the time‐accuracy at the coupled problem level. This feature is the main distinction between our framework and other approaches. To demonstrate the framework, we construct a family of partitioned heterogeneous time‐integration methods, combining multistage and multistep methods, for a simplified tracer transport component of the coupled air‐sea system in Earth system models. We report numerical tests evaluating accuracy and flux conservation for different pairs of time‐integrators from the explicit Runge‐Kutta and Adams‐Moulton families.

Computing generating sets of minimal size in finite algebras

We present an algorithm for calculating a minimal generating set of a finite algebra. Despite the fact that the problem is in NP, a single call to a SAT solver is impractical since the encoding is cubic. Instead, the proposed algorithm solves a series of smaller subproblems. The individual subproblems are formulated as integer linear programs (ILP) that are solved by an off-the-shelf solver. Our implementation shows that the proposed algorithm is highly efficient and is able to compute minimal generators for algebras of orders approximately 2000.In our experiments we focus on Moufang loops, a variety of loops with properties close to groups. For Moufang loops of prime power order, we are able to calculate a minimal generating set by another method, using theoretical results on the Frattini subloop and algorithms for permutation groups, of which some are reported here for the first time. This second method does not cover all cases, but in the covered cases it serves as a check of correctness of the ILP-based algorithm.

Magnetic Induction Tomography: Separation of the Ill-Posed and Non-Linear Inverse Problem into a Series of Isolated and Less Demanding Subproblems.

Magnetic induction tomography (MIT) is based on remotely excited eddy currents inside a measurement object. The conductivity distribution shapes the eddies, and their secondary fields are detected and used to reconstruct the conductivities. While the forward problem from given conductivities to detected signals can be unambiguously simulated, the inverse problem from received signals back to searched conductivities is a non-linear ill-posed problem that compromises MIT and results in rather blurry imaging. An MIT inversion is commonly applied over the entire process (i.e., localized conductivities are directly determined from specific signal features), but this involves considerable computation. The present more theoretical work treats the inverse problem as a non-retroactive series of four individual subproblems, each one less difficult by itself. The decoupled tasks yield better insights and control and promote more efficient computation. The overall problem is divided into an ill-posed but linear problem for reconstructing eddy currents from given signals and a nonlinear but benign problem for reconstructing conductivities from given eddies. The separated approach is unsuitable for common and circular MIT designs, as it merely fits the data structure of a recently presented and planar 3D MIT realization for large biomedical phantoms. For this MIT scanner, in discretization, the number of unknown and independent eddy current elements reflects the number of ultimately searched conductivities. For clarity and better representation, representative 2D bodies are used here and measured at the depth of the 3D scanner. The overall difficulty is not substantially smaller or different than for 3D bodies. In summary, the linear problem from signals to eddies dominates the overall MIT performance.

A Lagrangian Algorithm for Multiple Depot Traveling Salesman Problem With Revisit Period Constraints

This work presents a Multiple Depot Traveling Salesman Problem with revisit period constraints. The revisit period constraints are relevant to persistent routing applications, where these constraints represent maximum time between successive visits to a target. This problem is first posed as a Mixed Integer Linear Program. The coupling constraints in the primal problem are then relaxed via Lagrangian relaxation. Minimizing the resulting Lagrangian over the primal variables can be separated into an individual subproblem for each salesman. An algorithm to solve the subproblems to near-optimality that scales well for larger instances is presented. The dual function is then maximized, where three different dual update methods are studied: the subgradient method, the ellipsoid algorithm, and a bundle algorithm. Further, a primal reconstruction algorithm is presented to reconstruct feasible solutions from the solution of the dual algorithm. The quality of these solutions are compared to the optimal solutions obtained from the CPLEX MILP optimizer. The results of extensive numerical testing show that the dual algorithm presented was computationally efficient, capable of finding high quality solutions, and scale well compared to CPLEX. Further, it was seen that the bundle method used showed better convergence than the other update methods within the dual algorithm. Note to Practitioners— This work was motivated by the problem of routing UAVs in the presence of timing constraints, specifically motivated by military applications. This problem has constraints on the frequency at which targets are visited, such that high priority targets must be visited more frequently. Similar prior work exists in the literature for a variety of routing problems for different timing or resource constraints. Techniques which are successful on one set of constraints may be less useful when applied to different constraints. We present a Lagrangian based approach to the multiple depot traveling salesman variant. The results show the utility of a Lagrangian based technique to this problem, as both the solution quality and computation time scale well with problem size. The Lagrangian based technique presented here is seen to provide, on average, high quality solutions with improved computational time over a branch-and-bound algorithm, which solves the problem exactly. While the solutions returned from the algorithm are on average of good quality, they do exhibit some variance. Alternative algorithms may give more consistent results. The technique presented in this paper is general and may be applied to other routing problems where the revisit period constraints are applied. Future research includes application of this method to dynamic environments, where the targets’ states are unknown to the UAVs except when being monitored. Further, alternative techniques can be developed for this problem and compared to both the branch-and-bound and the Lagrange algorithm presented here.

Deep Learning for Commodity Procurement: Nonlinear Data-Driven Optimization of Hedging Decisions

As the number of exchange-traded commodity contracts and their volatility increase, risk management through financial hedging gains importance for commodity-purchasing firms. Existing data-driven optimization approaches for hedging decisions include linear regression-based techniques. As such, they assume linear price–feature relationships and, thus, do not automatically detect nonlinear feature effects. We propose an alternative, nonlinear data-driven approach to commodity procurement based on deep learning. The prescriptive algorithm uses artificial neural networks to allow for universal approximation and requires no a priori knowledge regarding underlying price processes. We reformulate the periodic review procurement problem as a multilabel time series classification problem as the optimal bang-bang type procurement policy allows us to treat the hedging decision for each demand period as an individual subproblem that is independent of the other periods. Thereby, we are differentiating between optimal and suboptimal hedging decisions in each period and introduce a novel opportunity cost–sensitive loss function. We train maximum likelihood classifiers based on different deep learning architectures and test their performance in numerical experiments and case studies for natural gas, crude oil, nickel, and copper procurement. We show comparable performance to the state of the art for linear price–feature relationships and considerable advantages in the nonlinear case. Funding: Financial support received through the DFG as part of the AdONE GRK2201 [Grant 277991500] is gratefully acknowledged.

A Review of Deep Learning-Based Visual Multi-Object Tracking Algorithms for Autonomous Driving

Multi-target tracking, a high-level vision job in computer vision, is crucial to understanding autonomous driving surroundings. Numerous top-notch multi-object tracking algorithms have evolved in recent years as a result of deep learning’s outstanding performance in the field of visual object tracking. There have been a number of evaluations on individual sub-problems, but none that cover the challenges, datasets, and algorithms associated with visual multi-object tracking in autonomous driving scenarios. In this research, we present an exhaustive study of algorithms in the field of visual multi-object tracking over the last ten years, based on a systematic review approach. The algorithm is broken down into three groups based on its structure: methods for tracking by detection (TBD), joint detection and tracking (JDT), and Transformer-based tracking. The research reveals that the TBD algorithm has a straightforward structure, however the correlation between its individual sub-modules is not very strong. To track multiple objects, the JDT technique combines multi-module joint learning with a deep network framework. Transformer-based algorithms have been explored over the past two years, and they have benefits in numerous assessment indicators, as well as tremendous research potential in the area of multi-object tracking. Theoretical support for algorithmic research in adjacent disciplines is provided by this paper. Additionally, the approach we discuss, which uses merely monocular cameras rather than sophisticated sensor fusion, is anticipated to pave the way for the quick creation of safe and affordable autonomous driving systems.

Energy Efficiency and Delay Tradeoff in an MEC-Enabled Mobile IoT Network

Mobile-edge computing (MEC) has recently emerged as a promising technology in the 5G era. It is deemed an effective paradigm to support computation intensive and delay-critical applications even at energy-constrained and computation-limited Internet of Things (IoT) devices. To effectively exploit the performance benefits enabled by MEC, it is imperative to jointly allocate radio and computational resources by considering nonstationary computation demands, user mobility, and wireless fading channels. This article aims to study the tradeoff between energy efficiency (EE) and service delay for multiuser multiserver MEC-enabled IoT systems when provisioning offloading services in a user mobility scenario. Particularly, we formulate a stochastic optimization problem with the objective of minimizing the long-term average network EE with the constraints of the task queue stability, peak transmit power, maximum CPU-cycle frequency, and maximum user number. To tackle the problem, we propose an online offloading and resource allocation algorithm by transforming the original problem into several individual subproblems in each time slot based on the Lyapunov optimization theory, which are then solved by convex decomposition and submodular methods. Theoretical analysis proves that the proposed algorithm can achieve a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[O(1/V), O(V)]$ </tex-math></inline-formula> tradeoff between EE and service delay. Simulation results verify the theoretical analysis and demonstrate our proposed algorithm can offer much better EE-delay performance in task offloading challenges, compared to several baselines.

Stage-t scenario dominance for risk-averse multi-stage stochastic mixed-integer programs

This paper presents a new and general approach, named “Stage-t Scenario Dominance,” to solve the risk-averse multi-stage stochastic mixed-integer programs (M-SMIPs). Given a monotonic objective function, our method derives a partial ordering of scenarios by pairwise comparing the realization of uncertain parameters at each time stage under each scenario. Specifically, we derive bounds and implications from the “Stage-t Scenario Dominance” by using the partial ordering of scenarios and solving a subset of individual scenario sub-problems up to stage t. Using these inferences, we generate new cutting planes to tackle the computational difficulty of risk-averse M-SMIPs. We also derive results on the minimum number of scenario-dominance relations generated. We demonstrate the use of this methodology on a stochastic version of the mean-conditional value-at-risk (CVaR) dynamic knapsack problem. Our computational experiments address those instances that have uncertainty, which correspond to the objective, left-hand side, and right-hand side parameters. Computational results show that our “scenario dominance"-based method can reduce the solution time for mean-risk, stochastic, multi-stage, and multi-dimensional knapsack problems with both integer and continuous variables, whose structure is similar to the mean-risk M-SMIPs, with varying risk characteristics by one-to-two orders of magnitude for varying numbers of random variables in each stage. Computational results also demonstrate that strong dominance cuts perform well for those instances with ten random variables in each stage, and ninety random variables in total. The proposed scenario dominance framework is general and can be applied to a wide range of risk-averse and risk-neutral M-SMIP problems.

A dynamic ship speed optimization method with time horizon segmentation

The determination of the appropriate operational profile of a ship through the adjustment of its sailing speed can potentially present a fruitful prospect in the effort towards the maximization of energy resources exploitation in commercial ships. The dependency of fuel consumption on speed, always in inseparable conjunction with the ever-changing weather conditions prevailing over any specified route, leads to the formulation of a (from mathematical point of view) not well-defined problem, when practical considerations regarding the capabilities of currently available weather forecast services are taken into account. In the present work, a method for the dynamic determination of optimal ship speed, in a fixed route, is proposed. Within this method, the problem of deteriorating accuracy of weather predictions for relatively long time periods is addressed with the segmentation of the route's total time horizon in smaller time periods, in order for solvable and meaningful optimization problems to be formulated. The interdependency between the individual sub-problems is established and examined with the definition of appropriate method parameters, which are representative of the most important time scales of the engineering problem at hand. The effectiveness of the method is demonstrated with the application on an actual container ship route.

Large-scale dynamic system optimization using dual decomposition method with approximate dynamic programming

In this paper, multi-agent dynamic optimization with a coupling constraint is studied. The aim is to minimize a strongly convex social cost function, by considering a linear stochastic dynamics for each agent and also coupling constraints among the agents. In order to handle the coupling constraint and also, to avoid high computational cost imposed by a centralized method for large scale systems, the dual decomposition method is used to decompose the problem into multiple individual sub-problems, while the dual variable is adjusted by a coordinator. Nevertheless, since each sub-problem is not a linear–quadratic (LQ) optimal control problem, and hence its closed-form solution does not exist, approximate dynamic programming (ADP) is utilized to solve the sub-problems. The main contribution of the paper is to propose an algorithm by considering the interrelated iterations of dual variable adjustment and ADP, and to prove the convergence of the algorithm to the global optimal solution of the social cost function. Additionally, the implementation of the proposed algorithm using a neural network is presented. Also, the computational advantage of the proposed algorithm in comparison with other bench-marking methods is discussed in simulation results.

Multi-UAV Enabled Data Collection with Efficient Joint Adaptive Interference Management and Trajectory Design

This paper studies interference in a data collection scenario in which multiple unmanned aerial vehicles (UAVs) are dispatched to wirelessly collect data from a set of distributed sensors. To improve the communication throughput and minimize the completion time, we design a joint resource allocation and trajectory optimization framework that not only is compatible with the traditional time-division scheme and interference coordination scheme but also combines their advantages. First, we analyse a basic quasi-stationary scenario with two UAVs and four devices, in which the two UAVs hover at optimal displacements to execute the data collection mission, and it is proven that the proposed optimal resource allocation and trajectory solution is adaptively adjustable according to the severity of the interference and that the common throughput of the network is non-decreasing. Second, for the general mobile case, we design an efficient algorithm to jointly address resource allocation and trajectory optimization, in which we first apply the block coordinate descent method to decompose the original non-convex problem into three non-convex sub-problems and then employ a dedicated genetic algorithm, a penalty function and the sequential convex approximation (SCA) technique to efficiently solve the individual sub-problems and obtain a satisfactory locally optimal solution with an adaptive initialization scheme. Subsequently, numerical experiments are presented to demonstrate that the completion time of the data collection task with our proposed method is at least 25% shorter than those with several baseline dynamic orthogonal schemes when 4 UAVs are deployed. Finally, we provide a practical application principle concerning the maximum suitable number of UAVs to avoid the inherent deficiencies of the proposed algorithm.

Dynamic Task Offloading and Resource Allocation for Mobile-Edge Computing in Dense Cloud RAN

With the unprecedented development of smart mobile devices (SMDs), e.g., Internet-of-Things devices and smartphones, various computation-intensive applications are explosively increasing in ultradense networks (UDNs). Mobile-edge computing (MEC) has emerged as a key technology to alleviate the computation workloads of SMDs and decrease service latency for computation-intensive applications. With the benefits of network function virtualization, MEC can be integrated with the cloud radio access network (C-RAN) in UDNs for computation and communication cooperation. However, with stochastic computation task arrivals and time-varying channel states, it is challenging to offload computation tasks online with energy-efficient computation and radio resource management. In this article, we investigate the task offloading and resource allocation problem in MEC-enabled dense C-RAN, aiming at optimizing network energy efficiency. A stochastic mixed-integer nonlinear programming problem is formulated to jointly optimize the task offloading decision, elastic computation resource scheduling, and radio resource allocation. To tackle the problem, the Lyapunov optimization theory is introduced to decompose the original problem into four individual subproblems which are solved by convex decomposition methods and matching game. We theoretically analyze the tradeoff between energy efficiency and service delay. Extensive simulations evaluate the impacts of system parameters on both energy efficiency and service delay. The simulation results also validate the superiority of the proposed task offloading and resource allocation scheme in dense C-RAN.

Similarity mapping for robust face recognition via a single training sample per person

Face recognition using a single training sample per person (STSPP) is very challenging due to significant variations in the query sample. Although most of sparse representation based methods eliminate the variations to obtain discriminative power, their recognition performance usually degrades as the number of training samples per person decreases. To improve recognition performance on the STSPP problem, in this paper, we propose a novel face recognition method: generating new training samples by preserving the variations of each query sample onto training samples instead of eliminating the variations, and then identifying the query one via the most similar generated training sample. Specifically, we formulate face recognition as an patch-based sparse representation problem, and then efficiently solve it by dividing the optimization procedure into three individual sub-problems: selecting training patches, weighting selected patches and calculating residuals. Meanwhile, we interpret the proposed algorithm from a probability modeling viewpoint. Experiments on multiple benchmark databases demonstrate its superior recognition performance to state-of-the-art methods on faces under illumination changes, occlusions and facial expressions, with lower test time costs.

Time-optimal control of large-scale systems of systems usingcompositional optimization

Optimization of industrial processes such as manufacturing cells canhave great impact on their performance. Finding optimal solutions to theselarge-scale systems is, however, a complex problem. They typically include multiplesubsystems, and the search space generally grows exponentially with each subsystem.In previous work we proposed CompositionalOptimization as a method to solve these type of problems. Thisintegrates optimization with techniques from compositional supervisory control,dividing the optimization into separate sub-problems. The main purpose is tomitigate the state explosion problem, but a bonusis that the individual sub-problems can be solved using parallel computation, makingthe method even more scalable. This paper further improves on compositionaloptimization with a novel synchronization method, called partial time-weighted synchronization (PTWS), that is specificallydesigned for time-optimal control of asynchronous systems. The benefit is itsability to combine the behaviour of asynchronous subsystems without introducingadditional states or transitions. The method also reduces the search space furtherby integrating an optimization heuristic that removes many non-optimal or redundantsolutions already during synchronization. Results in this paper show thatcompositional optimization efficiently generates global optimal solutions tolarge-scale realistic optimization problems, too big to solve when based ontraditional monolithic models. It is also shown that the introduction of PTWSdrastically decreases the total search space of the optimization compared toprevious work.

DeCODe: a community-based algorithm for generating high-quality decompositions of optimization problems

Nonconvex optimization problems, such as those often seen in chemical engineering applications due to integer decisions and inherent system physics, are known to scale poorly with problem size. Decomposition methods, such as Benders or Lagrangean decomposition, offer much promise for improving solution efficiency. However, automatically identifying subproblems for use with these solution methods remains an open problem. Herein, a general algorithmic framework entitled Detection of Communities for Optimization Decomposition (DeCODe) is presented. DeCODe uses community detection, a concept originating from network theory, to generate optimization subproblems which are strongly interacting within individual subproblems but weakly interacting between different ones. The importance of communities in solving problems using decomposition solution methods is showcased via a least squares regression problem. The ability of DeCODe to identify nontrivial decompositions of optimization problems is demonstrated through a large renewable energy and chemical production optimal design problem and two mixed integer nonlinear program test problems.