Augmented Lagrangian Relaxation Research Articles

Probabilistic Electricity Price Forecast With Optimal Prediction Interval

The uncertainty of electricity prices poses a challenge to all market participants as their decisions highly depend on the accuracy of price forecasts. Prediction intervals become an efficient method to quantify the uncertainties that reside in electricity price forecasts. In this paper, we propose a probabilistic electricity price forecast with optimal prediction interval method that considers both reliability and sharpness requirements. Taking reliability and sharpness into account, we ensure the prediction interval has a narrow width without sacrificing reliability. In the proposed method, the quantile regression is utilized to estimate the upper and lower bounds of the prediction intervals to avoid electricity price distribution assumption. In addition, the extreme learning machine (ELM) method is embedded in the forecast method to capture the nonlinear relationship within price data, and its tuning parameters are optimized using the augmented Lagrangian relaxation method in this paper. The effectiveness of the proposed probabilistic forecast method is demonstrated using data from various electricity markets.

Solving the multi-compartment vehicle routing problem by an augmented Lagrangian relaxation method

In real-world routing applications such as waste collection or fuel distribution, multiple products that cannot be mixed should be jointly collected or delivered. Multi-compartment vehicles are applied with great flexibility in routing and assignment decisions for these real-world applications. This study concentrates on the multi-compartment vehicle routing problem with fixed compartment size and fixed assignment products to compartments. For this problem, we develop two new formulations in the space–time network and state-space–time network. For the state-space–time network formulation, we dualize the task allocation constraints to the objective function and decompose the Lagrangian relaxed problem (LR) into a series of identical routing subproblems. To produce high-quality feasible solutions, the augmented Lagrangian relaxed problem (ALR) with superior convergence and feasibility is further applied to this problem. The ALR is decomposed into many solvable routing subproblems by alternating direction method of multipliers. A time-dependent dynamic programming algorithm is developed to settle the obtained subproblems of the LR and ALR. The subgradient optimization method solves the Lagrangian dual problems. The solution quality can be evaluated by the relative gap between the global lower and upper bounds at each iteration. In numerical experiments, the proposed method produces solutions with good integrality gaps. Based on Lagrangian relaxation and decomposition techniques, this research provides a novel solving scheme for the multi-compartment vehicle routing problem.

A distributed robust ADMM-based model for the energy management in local energy communities

Increasing the number of participants in energy communities leads to a new challenge in power systems, which is finding the optimal strategy for community members. Accordingly, this paper presents a distributed model for determining the optimal energy trading strategy of community participants such as buyers, sellers, and the community manager (CM). In the proposed model, the local day-ahead energy market, peer-to-peer (P2P) contracts, and the power grid are considered for trading energy between participants as well as compensating for power shortages/surpluses in the community. To model the uncertainty of PV generation, and selling/buying prices of the distribution network, the robust optimization (RO) approach is used. According to the defined budget of uncertainty, the optimal strategies of community members are determined based on the worst-case realizations of uncertain parameters. To decrease the solution time, the distributed optimization method is addressed. Accordingly, the augmented Lagrangian relaxation (ALR) and the alternating direction method of multipliers (ADMM) methods are used to decompose the optimization problem. The performance of the proposed model is evaluated through a case study. Simulation results demonstrate that the proposed model reduces the solution time of energy management problem in communities, significantly.

Joint location optimization of charging stations and segments in the space-time-electricity network: An augmented Lagrangian relaxation and ADMM-based decomposition scheme

Electric vehicles that contribute to better air quality, less noise, and low-carbon emissions are a promising selection for sustainable transportation. However, the development of electric vehicles is impeded by various factors, including driving range anxiety, long recharging period, and insufficient charging facilities. As the charging-while-driving techniques gradually mature, electric vehicles can recharge on charging stations stationarily or charging lanes (segments) dynamically. Additional charging facilities should be constructed to improve the level of charging services. Under a predefined construction budget, the difficulty is how to determine their numbers and distributions. This study jointly locates charging stations and segments by maximizing the accessibility of electric vehicles. In space-time-electricity networks, we formulate a multi-commodity network flow model with location and routing. A decomposition scheme based on augmented Lagrangian relaxation and alternating direction method of multipliers is developed to tackle this problem. The standard and augmented Lagrangian relaxed problems are decomposed into many solvable subproblems. Numerical experiments are conducted in three transportation networks, showing that the proposed method can achieve good integrality gaps.

Enhanced physics-informed neural networks with Augmented Lagrangian relaxation method (AL-PINNs)

Physics-Informed Neural Networks (PINNs) have become a prominent application of deep learning in scientific computation, as they are powerful approximators of solutions to nonlinear partial differential equations (PDEs). There have been numerous attempts to facilitate the training process of PINNs by adjusting the weight of each component of the loss function, called adaptive loss-balancing algorithms. In this paper, we propose an Augmented Lagrangian relaxation method for PINNs (AL-PINNs). We treat the initial and boundary conditions as constraints for the optimization problem of the PDE residual. By employing Augmented Lagrangian relaxation, the constrained optimization problem becomes a sequential max–min problem so that the learnable parameters λ adaptively balance each loss component. Our theoretical analysis reveals that the sequence of minimizers of the proposed loss functions converges to an actual solution for the Helmholtz, viscous Burgers, and Klein–Gordon equations. We demonstrate through various numerical experiments that AL-PINNs yield a much smaller relative error compared with that of state-of-the-art adaptive loss-balancing algorithms.

Subgroup fairness in two-sided markets.

It is well known that two-sided markets are unfair in a number of ways. For example, female drivers on ride-hailing platforms earn less than their male colleagues per mile driven. Similar observations have been made for other minority subgroups in other two-sided markets. Here, we suggest a novel market-clearing mechanism for two-sided markets, which promotes equalization of the pay per hour worked across multiple subgroups, as well as within each subgroup. In the process, we introduce a novel notion of subgroup fairness (which we call Inter-fairness), which can be combined with other notions of fairness within each subgroup (called Intra-fairness), and the utility for the customers (Customer-Care) in the objective of the market-clearing problem. Although the novel non-linear terms in the objective complicate market clearing by making the problem non-convex, we show that a certain non-convex augmented Lagrangian relaxation can be approximated to any precision in time polynomial in the number of market participants using semidefinite programming, thanks to its "hidden convexity". This makes it possible to implement the market-clearing mechanism efficiently. On the example of driver-ride assignment in an Uber-like system, we demonstrate the efficacy and scalability of the approach and trade-offs between Inter- and Intra-fairness.

Decentralized risk‐based security‐constrained optimal power flow in interconnected multi‐carrier microgrids

The security of energy systems has always been one of the most important issues of energy systems operation. The multi-carrier multi-microgrid (MCMMG) is one of the novel structures in the power system. High penetration of renewable resources and their uncertainty, their ability to operate in isolation mode, and the presence of energy hubs in MCMMG systems increase the importance of security assessment in this type of system. This paper presents a decentralized risk-based security-constrained optimal power flow model for an MCMMG system. The stochastic nature of wind energy, photovoltaic generation, and electrical and heating loads are considered in the proposed model. In the model, the risk level of the system is determined by considering both the electrical and natural gas (NG) networks’ contingencies. The proposed model is decomposed into three types of subproblems with the augmented Lagrangian relaxation (ALR) method: The subproblem related to the multi-microgrid operator level, the NG network subproblem, and the individual microgrid level subproblem. The decomposed problem is solved using MATLAB software. According to the results, the proposed model can secure the system from high costs that may impose on the system due to the contingency occurrence.

The Value of Preemptive Pick-Up Services in Dynamic Vehicle Routing for Last-Mile Delivery: Space-Time Network-Based Formulation and Solution Algorithms

In recent years, with the increase of emerging pick-up requests during service, logistics companies have been driven to integrate delivery and pick-up service in a dynamic environment. To provide a balanced and robust approach to cope with delivery requests and emerging pick-up requests, this article aims at considering and modeling a practically useful service principle as preemptive services. To our knowledge, most existing studies assume that the dynamically arriving requests are handled in a non-preemptive processing sequence; that is, once the delivery person is allocated to a task, the process is noninterruptible till it gets completed. In the preemptive service, a service suspension of the delivery process (with low service utility) is allowed to satisfy the pick-up requests (with high service utility) first. To provide a systematic assessment on the value of preemptive service for evolving urban logistics systems, a dynamic vehicle routing problem with preemptive pick-up service (VRPPS) is proposed to systematically describe the problem with potentially complex dynamic priorities among different tasks. Based on a dynamically constructed space-time network, this study formulates a multicommodity flow model that aims at optimizing the generalized service utility and the operating cost simultaneously. To provide a fast value approximation, we present a solution framework deploying the augmented Lagrangian relaxation approach with embedded dynamic programming algorithms. This framework jointly integrates the processes of updating request information and obtaining optimal routes. Finally, the validity and effectiveness of the proposed methods are evaluated on an illustrative network and a real-world last-mile delivery network operated by a logistics company.

A stochastic agent-based cooperative scheduling model of a multi-vector microgrid including electricity, hydrogen, and gas sectors

With increasing hydrogen usage, hydrogen subsystem should be considered in the multi-energy chain and a model is required that assumes current energy infrastructures, while preserving independent energy subsystems. In this study, a stochastic agent-based model is introduced for the coordinated scheduling of multi-vector microgrids considering interactions between electricity, hydrogen, and gas agents. Power to hydrogen (P2H) through electrolysis, hydrogen to power (H2P) through fuel cells, hydrogen to gas (H2G) through methanation, and gas to power (G2P) through distributed generation (DG) units are modeled to present the interactions among energy agents. The interactions in terms of shared variables and coupling constraints are described using augmented Lagrangian relaxation (ALR) and alternating direction method of multipliers (ADMM) to obtain three correlated optimization problems, preserving the privacy of energy sectors with minimum data exchange. An iterative process is accomplished among energy sectors to reach a consensus. Uncertainties in the wind turbine (WT) and photovoltaic (PV) power output, hydrogen vehicles (HVs), demands, and prices are captured using a stochastic method. To evaluate the proposed method, case studies are conducted using a multi-energy microgrid. The results verify that the microgrid is well scheduled and the interactions are accurately modeled, representing the effectiveness of the proposed method.

Centralized and Distributed Optimization for Vehicle-to-Grid Applications in Frequency Regulation

This paper proposes centralized and distributed optimization models for V2G applications to provide frequency regulation in power systems and the electricity market. Battery degradation and dynamic EV usages such as EV driving period, driving distance, and multiple charging/discharging locations are modeled. The centralized V2G problem is formulated into the linear programming (LP) model by introducing two sets of slack variables. However, the centralized model encounters limitations such as privacy concerns, high complexity, and central failure issues. To overcome these limitations, the distributed optimal V2G model is developed by decomposing the centralized model into subproblems using the augmented Lagrangian relaxation (ALR) method. The alternating direction method of multipliers (ADMM) is used to solve the distributed V2G model iteratively. The proposed models are evaluated using real data from the Independent Electricity System Operator (IESO) Ontario, Canada. Simulation results show that the proposed models can aggregate EVs for frequency regulation; meanwhile, the EV owners can obtain monetary rewards. The simulation also shows that including battery degradation and dynamic EV usage increases the model accuracy. By using the proposed approaches, the high cost and the low efficiency power generation units for frequency regulation can be compensated or partially replaced by EVs, which will reduce the generation cost and greenhouse gas emissions.

An augmented Lagrangian relaxation method for the mean-standard deviation based vehicle routing problem

With the ever-increasing trip demand and limited transportation resource, traffic congestions remain a tricky problem. Under traffic congestions, travel times are unstable and variable in a real-world network. When we conduct many time-sensitive vehicle routing optimizations such as vaccine transportation, it is essential to consider the travel time variability in these optimizations. However, most previous vehicle routing studies neglected the travel time variability and adopted the expected travel times or lengths of links. The historical travel times of multiple days allow us to access and describe the travel time variability. We can easily obtain the travel time means and standard deviations of links from these data. This study applies these two statistical indicators to vehicle routing optimization and concentrates on a mean-standard deviation based vehicle routing model. This model has a nonlinear and concave objective function that dramatically increases the problem complexity. We develop a novel augmented Lagrangian relaxation approach for this problem. A lower bound is constructed for this problem by applying the Lagrangian relaxation and decomposition technique twice. To break the inherent symmetry of this problem, we use the augmented Lagrangian relaxation method with block coordinate descent, linearization, and Lagrangian substitution techniques to achieve high-quality feasible solutions. Thus, the relative gap between the lower and upper bounds can be evaluated and reduced in the solving procedure. We implement the proposed method on the datasets modified from the standard benchmark instances. The average relative gaps of instances in sets A and P are 8.8% and 7.6%, respectively.

Equity‐oriented vehicle routing optimization for catering distribution services with timeliness requirements

The rapid development in the catering services and urban logistics industries has significantly promoted the prosperity of business-to-consumer (B2C) e-commerce urban logistics distribution, which is gaining increasing interest from food producers, distribution platforms, and consumers. Unlike traditional logistics supply chains, catering distribution services have strong timeliness and consideration of delivery delay. Traditional distribution platforms usually group commodities by their origins or destinations and then transport each group with one logistics vehicle. However, for a logistics distribution area with limited commodities, the vehicle capacity cannot be fully utilized if one vehicle can only transport commodities with the same origin or destination. Therefore, a mixed-load strategy is proposed in which commodities with different origins or destinations in a distribution area could be transported by the same vehicle to improve vehicle capacity utilization. A mixed-load strategy would further cause delivery sequencing problems, leading to different delivery delays for customers. This study proposed an equity-oriented vehicle routing problem for food distribution services with timeliness requirements considering a mixed-loading strategy and vehicle capacity constraints. For the above problem, a multi-commodity flow optimization model was constructed for the equity-oriented vehicle routing problem and a mixed-load strategy based on a time-discretized space-time-state network representation. An augmented Lagrangian relaxation approach was utilized to reformulate the original model and thus effectively solve the proposed model. Furthermore, the augmented Lagrangian model was decomposed and linearized into a series of shortest path searching subproblems and iteratively solved by a dynamic programming algorithm using an alternating direction method of multipliers (ADMM)-based solution framework. Finally, the proposed model and solution approaches were tested on numerous networks.

An Integer Mathematical Programming Model for Production Scheduling Problem in Open-Pit Mines under Grade Uncertainty and Solving Using the Firefly Algorithm

Long-term production scheduling in open-pit mines is a crucial issue in mining planning and determines the distribution of cash flow throughout the life of the mine. The purpose of the planning is to maximize the net present value by taking into account all operational constraints such as slope, mixing of different grades, mineral production, and extraction capacity. The uncertainties associated with model data play an important role in optimizing long-term production plans. Among the uncertainties, grade uncertainty plays a major role. In this paper, hybrid models are presented by the Lagrangian relaxation (LR) method, augmented Lagrangian relaxation (ALR) method, and firefly algorithm (FA) to solve the long-term production scheduling problem of open-pit mines with the assumption of deterministic and also considering the grade uncertainty. The firefly algorithm is used to update the Lagrange multipliers. The newly proposed approaches are based on optimizing Lagrangian multipliers and comparing them with the results of combined Lagrangian relaxation method and augmented Lagrangian relaxation with the Genetic Algorithm (GA), and the traditional sub-gradient (SG) method. For solving and validating the obtained model, Chadarmelo iron ore mine is considered as a suitable case study. The results of the case study show that the combined strategy (ALR-FA) can provide a near-optimal solution over other methods such that, over a given period, the net present value using the proposed hybrid approach is 20.11% higher than the traditional method is available. Also, the CPU speed of the proposed model is 4.7% more than the other methods.

Optimizing electric vehicle routing problems with mixed backhauls and recharging strategies in multi-dimensional representation network

Electric vehicles are environmental transportation modes that are widely applied in green logistics systems. To guarantee the energy efficiency, the impacts of customer service modes and recharging strategies need to be integrated into the optimization of electric logistics resource. This paper proposes an electric vehicle routing problem with mixed backhauls, time windows, and recharging strategies (EVRPMBTW-RS), minimizing the total travel cost with sophisticated constraints on the time-dependent pickup and delivery requests, limited recharging station capacity, and battery remaining capacity of electric vehicles. Mixed service sequences of linehaul and backhaul customers is allocated for the routing planning, with the synchronous optimization of recharging strategies including the selection of recharging stations and determination of recharging time. A time-discretized multi-commodity network flow model is constructed based on an extended space–time-state modeling framework, which is formulated as a quadratic 0–1 programming model by using the augmented Lagrangian relaxation technique. After the dualization and linearized transformation, we decompose the model into a sequence of least-cost path subproblems based on the alternating direction multiplier method (ADMM). The subproblems are alternately minimized and solved using the time-dependent forward dynamic programming algorithm. The solution quality can be guaranteed through calculating the optimality gap between the best lower bound and upper bound for each iteration. The proposed solution approach is examined on examples of a simple 7-node network and real-world Yizhuang road network. This paper provides a theoretical foundation for the route optimization method of electric logistics vehicles, and contributes to improve the operational efficiency of electric logistics systems.

A multi-block ADMM based approach for distribution market clearing with distribution locational marginal price

This work investigates a distribution locational marginal price (DLMP) scheme for promoting the market penetration of small-scale prosumers/consumers connected at distribution level and boosting their potential of demand response. In the distribution market, the distribution market operator (DMO) negotiates with each participant a node-specific DLMP and the respective demand profile of customer, so as to maximize the social welfare while respecting the operational constraints of the distribution system and load devices. First, different customers at demand side are modeled, including prosumers with photovoltaic (PV) generation, distributed storages and electric vehicles. All of them communicate with DMO and realize automated demand response by intelligent trading agent. Then, the distribution market clearing model for a massive number of customers in an ex-ante trade scheme is formulated as a market-based multi-period alternating current optimal power flow (ACOPF). As this clearing model is separable, the augmented Lagrangian relaxation (ALR) method is adopted to solve the market clearing model in a parallel manner. In the ALR, a second-order penalty term at each node is introduced to ensure the convergence, whereas it is not separable for customers. Hence, there are a multi-block demand variable coupled within the second-order penalty term at each node. A standard alternating direction method of multipliers (ADMM) algorithm cannot converge in this situation. To tackle this multi-block problem, we exploit the multi-block ADMM and distributed Dauglas-Rarchford Splitting method (d-DRSM), both of which proximally regularize the customer-specific subproblem to ensure global convergence. Furthermore, the prediction-correction mechanism in d-DRSM algorithm can achieve better rate of convergence in comparison with the multi-block ADMM. Finally, case studies are conducted and numerical results have shown the effectiveness of the proposed methodology, and the performance of different algorithms are analyzed for improving the clearing method’s efficiency.

Production scheduling problem and solver improvement via integration of the grey wolf optimizer into the augmented Lagrangian relaxation method

In open-pit mines, the long-term production scheduling problem is a mixed-integer programming optimization problem. Usually, it is tricky to reach the true optimal solution to the long-term production scheduling problem due to its complexity and dimensions. However, owing to its high-dimensional and combinatorial nature, the long-term production scheduling problem hinders the development of a precise mathematical optimization technique that can solve the whole problem for any real-size block model. In this paper, a hybrid model is introduced to solve the long-term production scheduling problem, taking into account grade uncertainty assisted by the augmented Lagrangian relaxation and the grey wolf optimizer methods. The hybrid augmented Lagrangian relaxation–the grey wolf optimizer technique is recommended to solve the long-term production scheduling problem to improve its performance and, consequently, speed up the convergence. The proposed model has been compared with the results of the hybrid methods gained from the classic Lagrangian relaxation and augmented Lagrangian relaxation methods integrated with the bat algorithm, particle swarm optimization, genetic algorithm, the traditional subgradient method, and conventional method without using the Lagrangian approach. The fallouts point out that better presentation is gained by the augmented Lagrangian relaxation–the grey wolf optimizer method in terms of net present value, average ore grade, and CPU time. Moreover, the cumulative net present value by the proposed model is 13.39% more than the conventional method.

Solving school bus routing problem with mixed-load allowance for multiple schools

The school bus routing problem (SBRP) is a challenging real-world problem that affects many citizens on a daily basis. This study considers several important classes of practical requirements for SBRP that include multiple schools, mixed-loads for students boarding the same bus but from different stops and commuting to different schools, heterogeneous fleets, various student-pickup time windows, and school-bell-ring constraints. Accordingly, a time-discretized multi-commodity network flow model is proposed based on a student-loading state-oriented space-time network. To enable optimization of both the student-bus assignment and bus routing, we introduce an extended-state dimension to represent the number of students commuting to different schools by buses. By implementing an augmented Lagrangian relaxation approach, the primal SBRP is reformulated as a quadratic 0–1 programming model with linear flow balance constraints. Furthermore, the augmented Lagrangian model can be decomposed and linearized as a series of linear multi-commodity network flow sub-problems that can be successively solved using dynamic programming algorithms in a cyclic block coordinate descent framework. The proposed model and approach to the solution are implemented on a nine-node test network, a Sioux Falls network under different scenarios, and a large-scale Chicago sketch network.

Integrated supply chain scheduling of procurement, production, and distribution under spillover effects

This paper considers an integrated scheduling problem in an e-commerce supply chain. The supply chain consists of heterogeneous suppliers, consumer-goods manufacturers that offer online channels, and a network of retailers. Suppliers provide raw materials or semi-finished products to manufacturers. Then, manufacturers produce and deliver end items to retailers. The retailers require that the manufacturers maintain a given fill rate, and, otherwise, a penalty will apply. Missing the fill rate can not only result in penalties, but also retailers’ switching, operational issues, and loss of image, and market value. Therefore, the total penalty costs are nonlinear with retailers’ unsatisfied quantities and the fill rate, which exhibits spillover effects. The problem is to select a subset of suppliers, assign retailers’ orders, and identify a production schedule with the manufacturers while minimizing the total costs that includes shipping and penalty. In this paper, we formulate a mixed-integer nonlinear program and develop a hybrid particle swarm optimization algorithm that can find a suitable solution within a reasonable time. The algorithm incorporates augmented Lagrangian relaxation and particle swarm optimization. We then perform computational testing on randomly generated cases and evaluate the performance of the proposed algorithm. Our numerical experiments show that the proposed algorithm is robust and efficient. Moreover, we provide a real case study, which demonstrates that the proposed model can lead to a substantial reduction of both the total and penalty costs.

Improving performance of open-pit mine production scheduling problem under grade uncertainty by hybrid algorithms

One of the surface mining methods is open-pit mining, by which a pit is dug to extract ore or waste downwards from the earth’s surface. In the mining industry, one of the most significant difficulties is long-term production scheduling (LTPS) of the open-pit mines. Deterministic and uncertainty-based approaches are identified as the main strategies, which have been widely used to cope with this problem. Within the last few years, many researchers have highly considered a new computational type, which is less costly, i.e., meta-heuristic methods, so as to solve the mine design and production scheduling problem. Although the optimality of the final solution cannot be guaranteed, they are able to produce sufficiently good solutions with relatively less computational costs. In the present paper, two hybrid models between augmented Lagrangian relaxation (ALR) and a particle swarm optimization (PSO) and ALR and bat algorithm (BA) are suggested so that the LTPS problem is solved under the condition of grade uncertainty. It is suggested to carry out the ALR method on the LTPS problem to improve its performance and accelerate the convergence. Moreover, the Lagrangian coefficients are updated by using PSO and BA. The presented models have been compared with the outcomes of the ALR-genetic algorithm, the ALR-traditional sub-gradient method, and the conventional method without using the Lagrangian approach. The results indicated that the ALR is considered a more efficient approach which can solve a large-scale problem and make a valid solution. Hence, it is more effectual than the conventional method. Furthermore, the time and cost of computation are diminished by the proposed hybrid strategies. The CPU time using the ALR-BA method is about 7.4% higher than the ALR-PSO approach.

A scenario-based hybrid robust and stochastic approach for joint planning of relief logistics and casualty distribution considering secondary disasters

This paper proposes a scenario-based three-stage hybrid robust and stochastic model that optimally designs the response network and distributes casualties effectively under uncertain combinational scenarios of primary and secondary disasters. Following the stochastic severity of combinational disasters, the robust counterparts are derived against the ambiguous uncertainty of evacuee scales and transportation time, respectively. A customized progressive hedging algorithm based on the augmented Lagrangian relaxation is developed to solve the problem. We decompose the problem based on the scenario and iteratively solve the adaptively penalized sub-problems with decision variables independent of stages. The results of an illustrative example show that incorporating secondary disaster scenarios can contribute to improving relief coverage. The proposed algorithm is competitive with some benchmarks.