Augmented Lagrangian Decomposition Research Articles

Coordinated Scheduling Strategy for Networked Microgrids Preserving Decision Independence and Information Privacy

With the increasing penetration of distributed renewable generations (DRGs), microgrids will play an important role in the future power system. This paper studies the coordinated scheduling strategy of networked microgrids with private data exchange limitations and local management independence. Based on an adaptive robust optimization method, a coordinated scheduling model of networked systems considering the uncertainty of renewable generations is established. Then distributed algorithms are developed to meet the needs of data privacy protection of individual microgrids. The Augmented Lagrangian (AL) decomposition method decomposes the model into several sub-problems, and an alternate optimization method is developed to speed up the solution. Case studies demonstrate the effectiveness of the proposed model and the solution methods.

Semi-proximal Augmented Lagrangian-Based Decomposition Methods for Primal Block-Angular Convex Composite Quadratic Conic Programming Problems

We first propose a semi-proximal augmented Lagrangian-based decomposition method to directly solve the primal form of a convex composite quadratic conic-programming problem with a primal block-angular structure. Using our algorithmic framework, we are able to naturally derive several well-known augmented Lagrangian-based decomposition methods for stochastic programming, such as the diagonal quadratic approximation method of Mulvey and Ruszczyński. Although it is natural to develop an augmented Lagrangian decomposition algorithm based on the primal problem, here, we demonstrate that it is, in fact, numerically more economical to solve the dual problem by an appropriately designed decomposition algorithm. In particular, we propose a semi-proximal symmetric Gauss–Seidel-based alternating direction method of multipliers (sGS-ADMM) for solving the corresponding dual problem. Numerical results show that our dual-based sGS-ADMM algorithm can very efficiently solve some very large instances of primal block-angular convex quadratic-programming problems. For example, one instance with more than 300,000 linear constraints and 12.5 million nonnegative variables is solved to the accuracy of 10-5 in the relative KKT residual in less than a minute on a modest desktop computer.

An Augmented Lagrangian Decomposition Method for Chance-Constrained Optimization Problems

Joint chance-constrained optimization problems under discrete distributions arise frequently in financial management and business operations. These problems can be reformulated as mixed-integer programs. The size of reformulated integer programs is usually very large even though the original problem is of medium size. This paper studies an augmented Lagrangian decomposition method for finding high-quality feasible solutions of complex optimization problems, including nonconvex chance-constrained problems. Different from the current augmented Lagrangian approaches, the proposed method allows randomness to appear in both the left-hand-side matrix and the right-hand-side vector of the chance constraint. In addition, the proposed method only requires solving a convex subproblem and a 0-1 knapsack subproblem at each iteration. Based on the special structure of the chance constraint, the 0-1 knapsack problem can be computed in quasi-linear time, which keeps the computation for discrete optimization subproblems at a relatively low level. The convergence of the method to a first-order stationary point is established under certain mild conditions. Numerical results are presented in comparison with a set of existing methods in the literature for various real-world models. It is observed that the proposed method compares favorably in terms of the quality of the best feasible solution obtained within a certain time for large-size problems, particularly when the objective function of the problem is nonconvex or the left-hand-side matrix of the constraints is random.

Resilient facility location against the risk of disruptions

In this paper, we consider an uncapacitated facility location problem (RUFL) with random facility disruptions. We develop risk-averse optimization formulations to compute resilient location and customer assignment solutions for two cases (i.e., under either independent or correlated disruptions), where the risks are expressed through a family of risk measures including conditional value-at-risk (CVaR) and absolute-semideviation (ASD). The risk-averse RUFL with independent facility disruptions is to control the risks at each individual customer and modeled as a mixed-integer nonlinear programming, which is challenging to be solved. In response, we develop a branch-and-cut algorithm combined with augmented Lagrangian decomposition for globally optimizing the problem. As for the risk-averse RUFL with correlated facility disruptions, we propose a scenario-based model to minimize the total fixed costs and risks across the entire customer set. The resulting formulation is a pure MILP and a Lagrangian decomposition scheme is proposed for computational aspects in large-scale cases. Our numerical results show that the risk-averse models outperform the classic risk-neutral models in improving the reliability. Experiments demonstrate that our proposed algorithms perform well. To conclude, we extract managerial insights that suggest important guidelines for controlling risk in the face of disruption.

Communication‐efficient distributed strategy for reactive power optimisation considering the uncertainty of renewable generation

Due to the non-linear power flow equality constraints and uncertain injected power of renewable generation, the reactive power optimisation problem is stochastic and non-convex, which is hard to be solved efficiently. Therefore, the discrete probability model of renewable generation is utilised to build the multi-scenario deterministic formulation of the stochastic problem, which is further changed to a conic optimisation model via the second-order conic relaxation. By using the skill of splitting buses and the augmented Lagrangian decomposition, the problem is separated into several smaller manageable sub-problems. Moreover, on the basis of the accelerated local augmented Lagrangian, a fully distributed conic optimisation algorithm is developed and the global optimal solution can be obtained iteratively in a parallel fashion. During the iterative procedure of the algorithm, only partial local message, without central coordination, need to be exchanged between the neighbouring subsystems. Each subsystem is interested in knowing only some components rather than the entire information, which improves communication efficiency. Simulations verify the effectiveness of the proposed algorithm.

Automated structure detection for distributed process optimization

The design and control of large-scale engineering systems, consisting of a number of interacting subsystems, is a heavily researched topic with relevance both for industry and academia. This paper presents two methodologies for optimal model-based decomposition, where an optimization problem is decomposed into several smaller sub-problems and subsequently solved by augmented Lagrangian decomposition methods. Large-scale and highly nonlinear problems commonly arise in process optimization, and could greatly benefit from these approaches, as they reduce the storage requirements and computational costs for global optimization. The strategy presented translates the problem into a constraint graph. The first approach uses a heuristic community detection algorithm to identify highly connected clusters in the optimization problem graph representation. The second approach uses a multilevel graph bisection algorithm to find the optimal partition, given a desired number of sub-problems. The partitioned graphs are translated back into decomposed sets of sub-problems with a minimal number of coupling constraints. Results show both of these methods can be used as efficient frameworks to decompose optimization problems in linear time, in comparison to traditional methods which require polynomial time.

Decentralized Multiarea Robust Generation Unit and Tie-Line Scheduling Under Wind Power Uncertainty

The growing interconnections of regional power systems and the large-scale integration of wind energy bring about the critical need to coordinate multiarea generation unit and tie-line scheduling (MAUTS). It is recognized that because of the limitations on private data exchange and model management, it is suitable to address the multiarea power scheduling problem in a decentralized way. In this paper, the MAUTS problem is formulated using the adaptive robust optimization (RO) scheme to account for uncertain wind energy. Our model is decomposed into regional subproblems by augmented Lagrangian decomposition (ALD), which enables a fully distributed computation within an alternating direction multiplier method framework. To address the nonconvexity issue, a tractable alternating optimization procedure (AOP) is developed to obtain high-quality solutions with finite convergence for the nonconvex mixed-integer problem. Simulations on different test systems are conducted to show the computational performance, the solution quality, and scalability of the proposed method.

An augmented Lagrangian method for distributed optimization

We propose a novel distributed method for convex optimization problems with a certain separability structure. The method is based on the augmented Lagrangian framework. We analyze its convergence and provide an application to two network models, as well as to a two-stage stochastic optimization problem. The proposed method compares favorably to two augmented Lagrangian decomposition methods known in the literature, as well as to decomposition methods based on the ordinary Lagrangian function.

An augmented Lagrangian decomposition approach for infrastructure maintenance and rehabilitation decisions under budget uncertainty

This article addresses a general class of infrastructure management problems raised in maintenance and rehabilitation scheduling. Infrastructure agencies usually face budget uncertainties that eventually lead to suboptimal scheduling planning if maintenance decisions are made without taking the uncertainty into consideration. It is important for decision makers to adopt maintenance and rehabilitation scheduling policies that consider all budget scenarios. The authors propose a multistage, stochastic programming model to address this problem. The authors also develop a solution procedure using the augmented Lagrangian decomposition algorithm. A case study exploring the computational characteristics of the proposed algorithm is conducted and the benefit ofusing the stochastic programming approach is discussed.

Distributed Supervisory Controller Design for Battery Swapping Modularity in Plug-In Hybrid Electric Vehicles

A distributed supervisory controller is proposed to achieve battery component swapping modularity (CSM) for a plug-in hybrid electric vehicle (PHEV). The CSM permits the designer to distribute a part of the supervisory controller to the battery module such that the PHEV can use a range of batteries while providing corresponding optimal fuel economy. A novel feedback-based controller for the charge sustaining mode is proposed to facilitate distributed controller design for battery CSM. The method based on sensitivity analysis of the control signals with respect to the battery hardware parameter is introduced to define the controller distribution architecture. The distributed controller gains are obtained by solving a bilevel optimization problem using the collaborative optimization and the augmented Lagrangian decomposition methods. The simulation results demonstrate that the proposed distributed controller can achieve battery CSM without compromising fuel economy compared to the centralized control case.

Integrated production planning and scheduling optimization of multisite, multiproduct process industry

The current manufacturing environment for process industry has changed from a traditional single-site, single market to a more integrated global production mode where multiple sites are serving a global market. In this paper, the integrated planning and scheduling problem for the multisite, multiproduct batch plants is considered. The major challenge for addressing this problem is that the corresponding optimization problem becomes computationally intractable as the number of production sites, markets, and products increases in the supply chain network. To effectively deal with the increasing complexity, the block angular structure of the constraints matrix is exploited by relaxing the inventory constraints between adjoining time periods using the augmented Lagrangian decomposition method. To resolve the issues of non-separable cross-product terms in the augmented Lagrangian function, we apply diagonal approximation method. Several examples have been studied to demonstrate that the proposed approach yields significant computational savings compared to the full-scale integrated model.

An Augmented Lagrangian Approach for Distributed Supply Chain Planning for Multiple Companies

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Planning coordination for multiple companies has received much attention from viewpoints of global supply chain management. In practical situations, a plausible plan for multiple companies should be created by mutual negotiation and coordination without sharing such confidential information as inventory costs, setup costs, and due date penalties for each company. In this paper, we propose a framework for distributed optimization of supply chain planning using an augmented Lagrangian decomposition and coordination approach. A feature of the proposed method is that it can derive a near-optimal solution without requiring all of the information. The proposed method is applied to supply chain planning problems for a petroleum complex, and a midterm planning problem for multiple companies. Computational experiments demonstrate that the average gap between a solution derived by the proposed method and the optimal solution is within 3% of the performance index, even though only local information is used to derive a solution for each company. </para>

限定的な情報共有環境下における拡張ラグランジュ分解調整法を適用した企業間交渉の最適化に関する研究

限定的な情報共有環境下における拡張ラグランジュ分解調整法を適用した企業間交渉の最適化に関する研究

Computational Assessment of Nested Benders and Augmented Lagrangian Decomposition for Mean-Variance Multistage Stochastic Problems

We consider decomposition approaches for the solution of multistage stochastic programs that appear in financial applications. In particular, we discuss the performance of two algorithms that we test on the mean-variance portfolio optimization problem. The first algorithm is based on a regularized version of Benders decomposition, and we discuss its extension to the quadratic case. The second algorithm is an augmented lagrangian method. Our results indicate that the algorithm based on regularized Benders decomposition is more efficient, which is in line with similar studies performed in the linear setting.

An augmented Lagrangian decomposition method for quasi-separable problems in MDO

Several decomposition methods have been proposed for the distributed optimal design of quasi-separable problems encountered in Multidisciplinary Design Optimization (MDO). Some of these methods are known to have numerical convergence difficulties that can be explained theoretically. We propose a new decomposition algorithm for quasi-separable MDO problems. In particular, we propose a decomposed problem formulation based on the augmented Lagrangian penalty function and the block coordinate descent algorithm. The proposed solution algorithm consists of inner and outer loops. In the outer loop, the augmented Lagrangian penalty parameters are updated. In the inner loop, our method alternates between solving an optimization master problem, and solving disciplinary optimization subproblems. The coordinating master problem can be solved analytically; the disciplinary subproblems can be solved using commonly available gradient-based optimization algorithms. The augmented Lagrangian decomposition method is derived such that existing proofs can be used to show convergence of the decomposition algorithm to KKT points of the original problem under mild assumptions. We investigate the numerical performance of the proposed method on two example problems.

A distributed decision making system for integrated optimization of production scheduling and distribution for aluminum production line

A distributed decision making system for integrated optimization of production scheduling and distribution planning is proposed. The system consists of several subsystems that individually derive the solution of its own subproblem to minimize the objective function obtained by decomposing the problem using an augmented Lagrangian decomposition and coordination technique. The proposed approach is applied to planning and scheduling for an aluminum rolling processing line. Decision variables such as arrival date of raw materials, lot-sizing, production scheduling, and allocation of products to warehouses are optimized simultaneously by repeating local optimization in material resource planning, and scheduling subsystem for each production process and warehouse planning subsystem. The effectiveness of the proposed approach is demonstrated.

Distributed route planning for multiple mobile robots using an augmented Lagrangian decomposition and coordination technique

To enable efficient transportation in semiconductor fabrication bays, it is necessary to generate route planning of multiple automated guided vehicles (AGVs) efficiently to minimize the total transportation time without collision among AGVs. In this paper, we propose a distributed route-planning method for multiple mobile robots using an augmented Lagrangian decomposition and coordination technique. The proposed method features a characteristic that each AGV individually creates a near-optimal routing plan through repetitive data exchange among the AGVs and local optimization for each AGV. Dijkstra's algorithm is used for local optimization. The optimality of the solution generated by the proposed method is evaluated by comparing the solution with an optimal solution derived by solving integer linear programming problems. A near-optimal solution, within 3% of the average gap from the optimal solution for an example transportation system consisting of 143 nodes and 14 AGVs, can be derived in less than 5 s of computation time for 100 types of requests. The proposed method is implemented in an experimental system with three AGVs, and the routing plan is derived in the configuration space, taking the motion of the robot into account. It is experimentally demonstrated that the proposed method is effective for various problems, despite the fact that each route for an AGV is created without minimizing the entire objective function.

An Augmented Lagrangian Approach for Scheduling Problems (Application to Total Weighted Tardiness Problems)

The paper describes an augmented Lagrangian decomposition and coordination approach for solving single machine scheduling problems to minimize the total weighted tardiness. The problem belongs to the class of NP-hard combinatorial optimization problem. We propose an augmented Lagrangian decomposition and coordination approach, which is commonly used for continuous optimization problems, for solving scheduling problems despite the fact that the problem is nonconvex and non-differentiable. The proposed method shows a good convergence to a feasible solution without heuristically constructing a feasible solution. The performance of the proposed method is compared with that of an ordinary Lagrangian relaxation.

拡張ラグランジュ分解調整法を用いた多企業間における入出荷計画問題の分散型最適化システム

拡張ラグランジュ分解調整法を用いた多企業間における入出荷計画問題の分散型最適化システム

Augmented Lagrangian Theory, Duality and Decomposition Methods for Variational Inequality Problems

In this paper, we develop the augmented Lagrangian theory and duality theory for variational inequality problems. We propose also decomposition methods based on the augmented Lagrangian for solving complex variational inequality problems with coupling constraints.