Lagrangian Relaxation for Mixed-Integer Linear Programming: Importance, Challenges, Recent Advancements, and Opportunities

Operations in areas of importance to society are frequently modeled as Mixed-Integer Linear Programming (MILP) problems. While MILP problems suffer from combinatorial complexity, Lagrangian Relaxation has been a beacon of hope to resolve the associated difficulties through decomposition. Due to the non-smooth nature of Lagrangian dual functions, the coordination aspect of the method has posed serious challenges. This paper presents several significant historical milestones (beginning with Polyak's pioneering work in 1967) toward improving Lagrangian Relaxation coordination through improved optimization of non-smooth functionals. Finally, this paper presents the most recent developments in Lagrangian Relaxation for fast resolution of MILP problems. The paper also briefly discusses the opportunities that Lagrangian Relaxation can provide at this point in time

Toward Efficient Transportation Electrification of Heavy-Duty Trucks: Joint Scheduling of Truck Routing and Charging

The timely transportation of goods to customers is an essential component of economic activities. However, heavy-duty diesel trucks used for goods delivery significantly contribute to greenhouse gas emissions within many large metropolitan areas, including Los Angeles, New York, and San Francisco. To reduce GHG emissions by facilitating freight electrification, this paper proposes Joint Routing and Charging scheduling for electric trucks. The objective of the associated optimization problem is to minimize the cost of transportation, charging, and tardiness. A large number of possible combinations of road segments as well as a large number of combinations of charging decisions and charging durations leads to a combinatorial explosion in the possible decisions electric trucks can make. The resulting mixed-integer linear programming problem is thus extremely challenging because of the combinatorial complexity even in the deterministic case. Therefore, a Surrogate Level-Based Lagrangian Relaxation (SLBLR) method is employed to decompose the overall problem into significantly less complex truck subproblems. In the coordination aspect, each truck subproblem is solved independently of other subproblems based on the values of Lagrangian multipliers. In addition to serving as a means of guiding and coordinating trucks, multipliers can also serve as a basis for transparent and explanatory decision-making by trucks. Testing results demonstrate that even small instances cannot be solved using the off-the-shelf solver CPLEX after several days of solving. The SLBLR method, on the other hand, can obtain near-optimal solutions within a few minutes for small cases, and within 30 minutes for large ones. Furthermore, it has been demonstrated that as battery capacity increases, the total cost decreases significantly; moreover, as the charging power increases, the number of trucks required decreases as well

Toward Robust Manufacturing Scheduling: Stochastic Job-Shop Scheduling

Manufacturing plays a significant role in promoting economic development, production, exports, and job creation, which ultimately contribute to improving the quality of life. The presence of manufacturing defects is, however, inevitable leading to products being discarded, i.e. scrapped. In some cases, defective products can be repaired through rework. Scrap and rework cause a longer completion time, which can contribute to the order being shipped late. In addition, complex manufacturing scheduling becomes much more challenging when the above uncertainties are present. Motivated by the presence of uncertainties as well as combinatorial complexity, this paper addresses the challenge illustrated through a case study of stochastic job-shop scheduling problems arising within low-volume high-variety manufacturing. To ensure on-time delivery, high-quality solutions are required, and near-optimal solutions must be obtained within strict time constraints to ensure smooth operations on the job-shop floor. To efficiently solve the stochastic job-shop scheduling (JSS) problem, a recently-developed Surrogate "Level-Based" Lagrangian Relaxation is used to reduce computational effort while efficiently exploiting the geometric convergence potential inherent to Polyak's step-sizing formula thereby leading to fast convergence. Numerical testing demonstrates that the new method is more than two orders of magnitude faster as compared to commercial solvers

Surrogate Lagrangian Relaxation: A Path To Retrain-free Deep Neural Network Pruning

Network pruning is a widely used technique to reduce computation cost and model size for deep neural networks. However, the typical three-stage pipeline significantly increases the overall training time. In this paper, we develop a systematic weight-pruning optimization approach based on Surrogate Lagrangian relaxation, which is tailored to overcome difficulties caused by the discrete nature of the weight-pruning problem. We prove that our method ensures fast convergence of the model compression problem, and the convergence of the SLR is accelerated by using quadratic penalties. Model parameters obtained by SLR during the training phase are much closer to their optimal values as compared to those obtained by other state-of-the-art methods. We evaluate our method on image classification tasks using CIFAR-10 and ImageNet with state-of-the-art MLP-Mixer, Swin Transformer, and VGG-16, ResNet-18, ResNet-50 and ResNet-110, MobileNetV2. We also evaluate object detection and segmentation tasks on COCO, KITTI benchmark, and TuSimple lane detection dataset using a variety of models. Experimental results demonstrate that our SLR-based weight-pruning optimization approach achieves a higher compression rate than state-of-the-art methods under the same accuracy requirement and also can achieve higher accuracy under the same compression rate requirement. Under classification tasks, our SLR approach converges to the desired accuracy $3\times$ faster on both of the datasets. Under object detection and segmentation tasks, SLR also converges $2\times$ faster to the desired accuracy. Further, our SLR achieves high model accuracy even at the hard-pruning stage without retraining, which reduces the traditional three-stage pruning into a two-stage process. Given a limited budget of retraining epochs, our approach quickly recovers the model's accuracy.Comment: arXiv admin note: text overlap with arXiv:2012.1007

Safety-assured, real-time neural active fault management for resilient microgrids integration

Federated-learning-based active fault management (AFM) is devised to achieve real-time safety assurance for microgrids and the main grid during faults. AFM was originally formulated as a distributed optimization problem. Here, federated learning is used to train each microgrid’s network with training data achieved from distributed optimization. The main contribution of this work is to replace the optimization-based AFM control algorithm with a learning-based AFM control algorithm. The replacement transfers computation from online to offline. With this replacement, the control algorithm can meet real-time requirements for a system with dozens of microgrids. By contrast, distributed-optimization-based fault management can output reference values fast enough for a system with several microgrids. More microgrids, however, lead to more computation time with optimization-based method. Distributed-optimization-based fault management would fail real-time requirements for a system with dozens of microgrids. Controller hardware-in-the-loop real-time simulations demonstrate that learning-based AFM can output reference values within 10 ms irrespective of the number of microgrids

Surrogate “Level-Based” Lagrangian Relaxation for mixed-integer linear programming

Abstract Mixed-Integer Linear Programming (MILP) plays an important role across a range of scientific disciplines and within areas of strategic importance to society. The MILP problems, however, suffer from combinatorial complexity. Because of integer decision variables, as the problem size increases, the number of possible solutions increases super-linearly thereby leading to a drastic increase in the computational effort. To efficiently solve MILP problems, a “price-based” decomposition and coordination approach is developed to exploit 1. the super-linear reduction of complexity upon the decomposition and 2. the geometric convergence potential inherent to Polyak’s stepsizing formula for the fastest coordination possible to obtain near-optimal solutions in a computationally efficient manner. Unlike all previous methods to set stepsizes heuristically by adjusting hyperparameters, the key novel way to obtain stepsizes is purely decision-based: a novel “auxiliary” constraint satisfaction problem is solved, from which the appropriate stepsizes are inferred. Testing results for large-scale Generalized Assignment Problems demonstrate that for the majority of instances, certifiably optimal solutions are obtained. For stochastic job-shop scheduling as well as for pharmaceutical scheduling, computational results demonstrate the two orders of magnitude speedup as compared to Branch-and-Cut. The new method has a major impact on the efficient resolution of complex Mixed-Integer Programming problems arising within a variety of scientific fields

Computationally Distributed and Asynchronous Operational Optimization of Droop-Controlled Networked Microgrids

Networked microgrids (MGs) with inverter-based and droop-controlled distributed energy resources (DERs) require operational optimization with guaranteed stability performance to ensure the stable energy supply with minimum cost, yet it remains an open challenge. Additionally, the discrete nature of MGs leads to convergence issues to existing optimization methods thereby leading to difficulties obtaining feasible solutions for large-scale networks. This article develops a paradigm for discrete droop control to improve microgrids’ controllability in managing voltage and frequency fluctuations. With the emergence of Internet of Things, the computational tasks are distributed among local resources. The utilized Distributed and Asynchronous Surrogate Lagrangian Relaxation (DA-SLR) method distributes the optimization tasks among the MGs and efficiently coordinates the distributed subsystems. A small-signal model of the operational optimization is then developed to verify the system’s stability. Numerous case studies have proven the DA-SLR’s efficacy in comparison to various variations of the alternating direction of multipliers method (ADMM)

Efficient Operations of Micro-Grids with Meshed Topology and Under Uncertainty through Exact Satisfaction of AC-PF, Droop Control and Tap-Changer Constraints

Micro-grids’ operations offer local reliability; in the event of faults or low voltage/frequency events on the utility side, micro-grids can disconnect from the main grid and operate autonomously while providing a continued supply of power to local customers. With the ever-increasing penetration of renewable generation, however, operations of micro-grids become increasingly complicated because of the associated fluctuations of voltages. As a result, transformer taps are adjusted frequently, thereby leading to fast degradation of expensive tap-changer transformers. In the islanding mode, the difficulties also come from the drop in voltage and frequency upon disconnecting from the main grid. To appropriately model the above, non-linear AC power flow constraints are necessary. Computationally, the discrete nature of tap-changer operations and the stochasticity caused by renewables add two layers of difficulty on top of a complicated AC-OPF problem. To resolve the above computational difficulties, the main principles of the recently developed “l1-proximal” Surrogate Lagrangian Relaxation are extended. Testing results based on the nine-bus system demonstrate the efficiency of the method to obtain the exact feasible solutions for micro-grid operations, thereby avoiding approximations inherent to existing methods; in particular, fast convergence of the method to feasible solutions is demonstrated. It is also demonstrated that through the optimization, the number of tap changes is drastically reduced, and the method is capable of efficiently handling networks with meshed topologies

Efficient Operations of Micro-Grids with Meshed Topology and Under Uncertainty through Exact Satisfaction of AC-PF, Droop Control and Tap-Changer Constraints

Micro-grids’ operations offer local reliability; in the event of faults or low voltage/frequency events on the utility side, micro-grids can disconnect from the main grid and operate autonomously while providing a continued supply of power to local customers. With the ever-increasing penetration of renewable generation, however, operations of micro-grids become increasingly complicated because of the associated fluctuations of voltages. As a result, transformer taps are adjusted frequently, thereby leading to fast degradation of expensive tap-changer transformers. In the islanding mode, the difficulties also come from the drop in voltage and frequency upon disconnecting from the main grid. To appropriately model the above, non-linear AC power flow constraints are necessary. Computationally, the discrete nature of tap-changer operations and the stochasticity caused by renewables add two layers of difficulty on top of a complicated AC-OPF problem. To resolve the above computational difficulties, the main principles of the recently developed “l1-proximal” Surrogate Lagrangian Relaxation are extended. Testing results based on the nine-bus system demonstrate the efficiency of the method to obtain the exact feasible solutions for micro-grid operations, thereby avoiding approximations inherent to existing methods; in particular, fast convergence of the method to feasible solutions is demonstrated. It is also demonstrated that through the optimization, the number of tap changes is drastically reduced, and the method is capable of efficiently handling networks with meshed topologies