Augmented Lagrangian Coordination Research Articles

Multidisciplinary Design Optimization Approach to Integrated Space Mission Planning and Spacecraft Design

Space mission planning and spacecraft design are tightly coupled and need to be considered together for optimal performance; however, this integrated optimization problem results in a large-scale mixed-integer nonlinear programming (MINLP) problem, which is challenging to solve. In response to this challenge, this paper proposes a new solution approach to this problem based on decomposition-based optimization via augmented Lagrangian coordination. The proposed approach leverages the unique structure of the problem that enables its decomposition into a set of coupled subproblems of different types: a mixed-integer quadratic programming (MIQP) subproblem for mission planning, and one or more nonlinear programming (NLP) subproblem(s) for spacecraft design. Because specialized MIQP or NLP solvers can be applied to each subproblem, the proposed approach can efficiently solve the otherwise intractable integrated MINLP problem. An automatic and effective method to find an initial solution for this iterative approach is also proposed so that the optimization can be performed without a user-defined initial guess. The demonstration case study shows that, compared to the state-of-the-art method, the proposed formulation converges substantially faster and the converged solution is at least the same or better given the same computational time limit.

Augmented Lagrangian coordination for energy-optimal allocation of smart manufacturing services

• Energy-optimal service allocation model in smart manufacturing is constructed. • The diverse production domains of manufacturing tasks are considered. • A distributed optimization method is adopted to solve the proposed problem. • The performance of the proposed method is verified by a case study. The rapid development of information and communication technologies has triggered the proposition and implementation of smart manufacturing paradigms. In this regard, efficient allocation of smart manufacturing services (SMSs) can provide a sustainable manner for promoting cleaner production. Currently, centralized optimization methods have been widely used to complete the optimal allocation of SMSs. However, personalized manufacturing tasks usually belong to diverse production domains. The centralized optimization methods could hardly include related production knowledge of all manufacturing tasks in an individual decision model. Consequently, it is difficult to provide satisfactory SMSs for meeting customer's requirements. In addition, energy consumption is rarely considered in the SMS allocation process which is unfavorable for performing sustainable manufacturing. To address these challenges, augmented Lagrangian coordination (ALC), a novel distributed optimization method is proposed to deal with the energy-optimal SMS allocation problem in this paper. The energy-optimal SMS allocation model is constructed and decomposed into several loose-coupled and distributed elements. Two variants of the ALC method are implemented to formulate the proposed problem and obtain final SMS allocation results. A case study is employed to verify the superiority of the proposed method in dealing with energy-optimal SMS allocation problems by comparing with the centralized optimization method at last.

Probabilistic analytical target cascading using kernel density estimation for accurate uncertainty propagation

Probabilistic analytical target cascading (PATC) has been developed to incorporate uncertainty of random variables in a hierarchical multilevel system using the framework of ATC. In the decomposed ATC structure, consistency between linked subsystems has to be guaranteed through individual subsystem optimizations employing special coordination strategies such as augmented Lagrangian coordination (ALC). However, the consistency in PATC has to be attained exploiting uncertainty quantification and propagation of interrelated linking variables that are the major concern of PATC and uncertainty-based multidisciplinary design optimization (UMDO). In previous studies, the consistency of linking variables is assured by matching statistical moments under the normality assumption. However, it can induce significant error when the linking variable to be quantified is highly nonlinear and non-normal. In addition, reliability estimated from statistical moments may be inaccurate in each optimization of the subsystem. To tackle the challenges, we propose the sampling-based PATC using multivariate kernel density estimation (KDE). The framework of reliability-based design optimization (RBDO) using sampling methods is adopted in individual optimizations of subsystems in the presence of uncertainty. The uncertainty quantification of linking variables equivalent to intermediate random responses can be achieved by multivariate KDE to account for correlation between linking variables. The constructed KDE based on finite samples of the linking variables can provide accurate statistical representations to linked subsystems and thus be utilized as probability density function (PDF) of linking variables in individual sampling-based RBDOs. Stochastic sensitivity analysis with respect to multivariate KDE is further developed to provide an accurate sensitivity of reliability during the RBDO. The proposed sampling-based PATC using KDE facilitates efficient and accurate procedures to obtain a system optimum in PATC, and the mathematical examples and roof assembly optimization using finite element analysis (FEA) are used to demonstrate the effectiveness of the proposed approach.

Improved augmented Lagrangian coordination for optimizing supply chain configuration with multiple sharing elements in industrial cluster

PurposeThe purpose of this paper is to study various combination forms of the three basic sharing elements (i.e. orders sharing, manufacturers capacity sharing and suppliers capacity sharing) in the cluster supply chain (CSC), formulate a distributed model to protect enterprises’ decision privacy and seek to develop an effective method for solving the distributed complex model.Design/methodology/approachA distributed assembly cluster supply chain configuration (ACSCC) model is formulated. An improved augmented Lagrangian coordination (ALC) is proposed and used to solve the ACSCC model. A series of experiments are conducted to validate the improved ALC and the model.FindingsTwo major findings are obtained. First, the market order’s quantity change and the sales price of the product have a great impact on both the optimal results of the ACSCC and the cooperative strategy, especially, when the market order increases sharply, enterprises have to adopt multiple cooperative strategies to complete the order; meanwhile, the lower sales price of the product helps independent suppliers to get more orders. Second, the efficiency and computational accuracy of the improved ALC method are validated as compared to the centralized ALC and Lingo11.Research limitations/implicationsThis paper formulated the single-period ACSCC model under certain assumptions, yet a multi-period ACSCC model is to be developed, a more comprehensive investigation of the relationships among combination forms is to be extended further and a rigid proof of the improved ALC is necessary.Practical implicationsEnterprises in the industrial cluster should adopt different cooperative strategies in terms of the market order’s quantity change and the sales price of the product.Social implicationsThe proposed various combination forms of sharing elements and the formulated ACSCC model provide guidance to managers in the industrial cluster to choose the proper policy.Originality/valueThis research studies various combination forms of the three basic sharing elements in the CSC. A distributed ACSCC model has been established considering simultaneously multiple sharing elements. An improved ALC is presented and applied to the ACSCC problem.

Extending augmented Lagrangian coordination for the optimal configuration of cloud-based smart manufacturing services with production capacity constraint

Abstract Cloud manufacturing (CMfg) has been widely recognized since production resources could be encapsulated into smart manufacturing services and managed by advanced information and manufacturing technologies. As a core part for promoting sustainable production processes, manufacturing service configuration (MSC) aims to optimize the allocation of services for tasks in CMfg. This research studies an MSC problem considering the decision autonomy (DA) and limited production capacities (PC) of service providers (MSC-DA-PC). Augmented Lagrangian coordination (ALC), an emerged distributed optimization method, can support open-structure collaboration and allow participants to maintain decision autonomy. In this paper, ALC is extended to solve the proposed MSC-DA-PC problem by the introduction of a novel coordination element (“CO” element). The working logic and solution strategy of the “CO” element are investigated. Two case studies are employed to verify the effectiveness and efficiency of the extended ALC method. The observation shows the dynamic formation of MSC results along with the changing of task quantity.

Modified augmented Lagrangian coordination and alternating direction method of multipliers with parallelization in non-hierarchical analytical target cascading

Analytical Target Cascading (ATC) is a decomposition-based optimization methodology that partitions a system into subsystems and then coordinates targets and responses among subsystems. Augmented Lagrangian with Alternating Direction method of multipliers (AL-AD), one of efficient ATC coordination methods, has been widely used in both hierarchical and non-hierarchical ATC and theoretically guarantees convergence under the assumption that all subsystem problems are convex and continuous. One of the main advantages of distributed coordination which consists of several non-hierarchical subproblems is that it can solve subsystem problems in parallel and thus reduce computational time. Therefore, previous studies have proposed an augmented Lagrangian coordination strategy for parallelization by eliminating interactions among subproblems. The parallelization is achieved by introducing a master problem and support variables or by approximating a quadratic penalty function to make subproblems separable. However, conventional AL-AD does not guarantee convergence in the case of parallel solving. Our study shows that, in parallel solving using targets and responses of the current iteration, conventional AL-AD causes mismatch of information in updating the Lagrange multiplier. Therefore, the Lagrange multiplier may not reach the optimal point, and as a result, increasing penalty weight causes numerical difficulty in the augmented Lagrangian coordination approach. To solve this problem, we propose a modified AL-AD with parallelization in non-hierarchical ATC. The proposed algorithm uses the subgradient method with adaptive step size in updating the Lagrange multiplier and also maintains penalty weight at an appropriate level not to cause oscillation. Without approximation or introduction of an artificial master problem, the modified AL-AD with parallelization can achieve similar accuracy and convergence with much less computational cost compared with conventional AL-AD with sequential solving.

An augmented Lagrangian coordination method for optimal allocation of cloud manufacturing services

With the rapid development of information and manufacturing technologies, cloud manufacturing (CMfg) was proposed and attracted wide attention. In CMfg, manufacturing service allocation (MSA) plays an important role in facilitating high-quality service management. MSA aims to optimize the allocation of services for manufacturing tasks. All-in-one (AIO) methods are widely used to obtain an optimal MSA result. However, the current AIO methods usually use one decision model so that it is difficult to maintain the autonomous decision rights of service providers. As a distributed optimization mechanism, augmented Lagrangian coordination (ALC) can offer an open-structure collaboration and allow participants to keep autonomous decision rights. In this paper, the MSA problem is partitioned into an ALC model based on the decision scope of service providers and solved in a loose coupling and distributed manner. A case study demonstrates the specific steps of ALC for solving the MSA problem. The results show the effectiveness and efficiency of ALC method in solving the MSA problem, as well as its promising feature in maintaining decision autonomy of a service provider.

Optimal configuration of assembly supply chains based on Hybrid augmented Lagrangian coordination in an industrial cluster

Industrial cluster is becoming an ever more important cost-effective industry development mode especially when enterprises are required to give more rapid responses to the frequently changed customized demands. The explosive number of homogeneous enterprises/suppliers with geographic proximity provides multiple options for each supply chain stage, which thus leads to higher potential to form a more satisfactorily performed assembly supply chain (assembly system) in industrial clusters. However, the increased candidate options also incur inevitably higher decision complexity to the decision model of configuring such cluster supply chains. The situation may be more challenging if the autonomous decision requirement of individual suppliers is accommodated. A general assembly cluster supply chain configuration (ACSCC) model is established which considers both horizontally and vertically collaborations in a cluster, meaning it accommodates the typical cluster relationships including subcontracting and outsourcing. In order to achieve the complexity reduction and autonomy protection, a newly emerged decomposition-based solution method named augmented Lagrangian coordination (ALC) will be adopted. Especially, two classical ALC formulation variants named the centralized coordination formulation and the distributed coordination formulation are innovatively integrated to form a hybrid ALC solution strategy, which deals with different assembly branches with different alliancing structures. Experimental results have proved the effectiveness of the proposed hybrid ALC method for the ACSCC problem. From the perspective of supply chain management, a set of sensitivity analysis for profit of each collaborative enterprise is conducted to obtain some important managerial insights.

Improving the Performance of the Augmented Lagrangian Coordination: Decomposition Variants and Dual Residuals

The augmented Lagrangian coordination (ALC), as an effective coordination method for decomposition-based optimization, offers significant flexibility by providing different variants when solving nonhierarchically decomposed problems. In this paper, these ALC variants are analyzed with respect to the number of levels and multipliers, and the resulting advantages and disadvantages are explored through numerical tests. The efficiency, accuracy, and parallelism of three ALC variants (distributed ALC, centralized ALC, and analytical target cascading (ATC) extended by ALC) are discussed and compared. Furthermore, the dual residual theory for the centralized ALC is extended to the distributed ALC, and a new flexible nonmonotone weight update is proposed and tested. Numerical tests show that the proposed update effectively improves the accuracy and robustness of the distributed ALC on a benchmark engineering test problem.

Dual Residual for Centralized Augmented Lagrangian Coordination Based on Optimality Conditions

Centralized augmented Lagrangian coordination (ALC) has drawn much attention due to its parallel computation capability, efficiency, and flexibility. The initial setting and update strategy of the penalty weights in this method are critical to its performance. The traditional weight update strategy always increases the weights and research shows that inappropriate initial weights may cause optimization failure. Making use of the Karush–Kuhn–Tucker (KKT) optimality conditions for the all-in-one (AIO) and decomposed problems, the terms “primal residual” and “dual residual” are introduced into the centralized ALC, and a new update strategy considering both residuals and thus guaranteeing the unmet optimality condition in the traditional update is introduced. Numerical tests show a decrease in the iteration number and significant improvements in solution accuracy with both calculated and fine-tuned initial weights using the new update. Additionally, the proposed approach is capable to start from a wide range of possible weights and achieve optimality, and therefore brings robustness to the centralized ALC.

Optimal configuration of cluster supply chains with augmented Lagrange coordination

Enterprises in an industrial cluster could dynamically alliance in the form of cluster supply chains to share inner-cluster resources and services, and respond to the ever-fluctuating customer demands in a cost-effective way. However, an effective and feasible method enabling such dynamic cluster supply chain configuration (CSCC) lags behind practice due to the conflict of interests. Researchers are designing All-in-One theoretic models to optimize CSCC with the assumed decision details of all enterprises, while in fact clustered enterprises are seeking effective decentralized decision mechanisms which protect their decision autonomy in the frequently re-configured CSC. A newly emerged multi-disciplinary optimization method, Augmented Lagrangian Coordination (ALC), which supports the open-structure collaboration with strict optimization convergence, is thoroughly investigated in this paper and applied to solve the conflict. Through a complete analysis of CSC’s configuration policies in typical stages, a generic CSCC model is proposed and then partitioned into an ALC-based decentralized decision model by the typical decision autonomy distribution in clusters. Clustered enterprises collaborate vertically and laterally along the ALC model through multi-dimensional couplings to achieve the overall consistency and optimality. Results have proved the effectiveness of ALC for CSCC problem. A set of sensitivity analysis is also conducted to find out the condition in which an order has to be fulfilled in a CSC and the most appropriate configuration.

Distributed Computation of Pareto Sets

The needs of multidisciplinary engineering design have motivated the development of distributed solution approaches to computing efficient solutions to decomposable multiobjective optimization problems (MOPs). The decomposition is necessary due to the assumption that the overall MOP is not solvable since access to its solution space is subproblem-restricted. The state-of-the-art analyses for distributed solution approaches such as the alternating direction method of multipliers (ADMM) and the block coordinate descent (BCD) method were developed in the context of problem decomposition originating in a single objective setting and are not immediately applicable to MOPs. Applying certain scalarization techniques well-suited for nonconvex MOPs, the decomposable MOP is reformulated into a single objective problem (SOP) but the decomposability is not preserved and the SOP is not suitable for the application of ADMM. Furthermore, coupling between the subproblems makes BCD in its current form likewise inadequate. To address these challenges to distributed multiobjective optimization, existing theory is extended for (1) iterative augmented Lagrangian coordination techniques and (2) the block coordinate descent method. For the former, each augmented Lagrangian subproblem is subject to convex constraints and is solved approximately for solutions satisfying a necessary condition for optimality (but not necessarily optimal). For the latter, existing convergence proofs under generalized convexity assumptions are redeveloped so that new insights and convergence results become evident. An integration of the above two convergence analyses yields convergence results for a distributed solution approach that addresses coupling between subproblems and does not require convergence of the generated sequence of solution-multiplier pairs. Based on this study, a multi-objective decomposition algorithm is developed for the distributed generation of efficient solutions to nonconvex decomposable MOPs.

Cutting Plane Methods for Analytical Target Cascading With Augmented Lagrangian Coordination

Analytical target cascading (ATC), a hierarchical, multilevel, multidisciplinary coordination method, has proven to be an effective decomposition approach for large-scale engineering optimization problems. In recent years, augmented Lagrangian relaxation methods have received renewed interest as dual update methods for solving ATC decomposed problems. These problems can be solved using the subgradient optimization algorithm, the application of which includes three schemes for updating dual variables. To address the convergence efficiency disadvantages of the existing dual update schemes, this paper investigates two new schemes, the linear and the proximal cutting plane methods, which are implemented in conjunction with augmented Lagrangian coordination for ATC-decomposed problems. Three nonconvex nonlinear example problems are used to show that these two cutting plane methods can significantly reduce the number of iterations and the number of function evaluations when compared to the traditional subgradient update methods. In addition, these methods are also compared to the method of multipliers and its variants, showing similar performance.

Consistency Constraint Allocation in Augmented Lagrangian Coordination

Many engineering systems are too complex to design as a single entity. Decomposition-based design optimization methods partition a system design problem into subproblems, and coordinate subproblem solutions toward an optimal system design. Recent work has addressed formal methods for determining an ideal system partition and coordination strategy, but coordination decisions have been limited to subproblem sequencing. An additional element in a coordination strategy is the linking structure of the partitioned problem, i.e., the allocation of constraints that guarantee that the linking variables among subproblems are consistent. There may exist many alternative linking structures for a decomposition-based strategy that can be selected for a given partition, and this selection should be part of an optimal simultaneous partitioning and coordination scheme. This article develops a linking structure theory for a particular class of decomposition-based optimization algorithms, augmented Lagrangian coordination (ALC). A new formulation and coordination technique for parallel ALC implementations is introduced along with a specific linking structure theory, yielding a partitioning and coordination selection method for ALC that includes consistency constraint allocation. This method is demonstrated using an electric water pump design problem.

A Nonhierarchical Formulation of Analytical Target Cascading

Analytical target cascading (ATC) is a method developed originally for translating system-level design targets to design specifications for the components that comprise the system. ATC has been shown to be useful for coordinating decomposition-based optimal system design. The traditional ATC formulation uses hierarchical problem decompositions, in which coordination is performed by communicating target and response values between parents and children. The hierarchical formulation may not be suitable for general multidisciplinary design optimization (MDO) problems. This paper presents a new ATC formulation that allows nonhierarchical target-response coupling between subproblems and introduces system-wide functions that depend on variables of two or more subproblems. Options to parallelize the subproblem optimizations are also provided, including a new bilevel coordination strategy that uses a master problem formulation. The new formulation increases the applicability of the ATC to both decomposition-based optimal system design and MDO. Moreover, it belongs to the class of augmented Lagrangian coordination methods, having thus convergence properties under standard convexity and continuity assumptions. A supersonic business jet design problem is used to demonstrate the flexibility and effectiveness of the presented formulation.

A micro-accelerometer MDO benchmark problem

Many optimization and coordination methods for multidisciplinary design optimization (MDO) have been proposed in the last three decades. Suitable MDO benchmark problems for testing and comparing these methods are few however. This article presents a new MDO benchmark problem based on the design optimization of an ADXL150 type lateral capacitive micro-accelerometer. The behavioral models describe structural and dynamic effects, as well as electrostatic and amplification circuit contributions. Models for important performance indicators such as sensitivity, range, noise, and footprint area are presented. Geometric and functional constraints are included in these models to enforce proper functioning of the device. The developed models are analytical, and therefore highly suitable for benchmark and educational purposes. Four different problem decompositions are suggested for four design cases, each of which can be used for testing MDO coordination algorithms. As a reference, results for an all-in-one implementation, and a number of augmented Lagrangian coordination algorithms are given.

Multi-modality in augmented Lagrangian coordination for distributed optimal design

This paper presents an empirical study of the convergence characteristics of augmented Lagrangian coordination (ALC) for solving multi-modal optimization problems in a distributed fashion. A number of test problems that do not satisfy all assumptions of the convergence proof for ALC are selected to demonstrate the convergence characteristics of ALC algorithms. When only a local search is employed at the subproblems, local solutions to the original problem are often attained. When a global search is performed at subproblems, global solutions to the original, non-decomposed problem are found for many of the examples. Although these findings are promising, ALC with a global subproblem search may yield only local solutions in the case of non-convex coupling functions or disconnected feasible domains. Results indicate that for these examples both the starting point and the sequence in which subproblems are solved determines which solution is obtained. We illustrate that the main cause for this behavior lies in the alternating minimization inner loop, which is inherently of a local nature.

Block-separable linking constraints in augmented Lagrangian coordination

Augmented Lagrangian coordination (ALC) is a provably convergent coordination method for multidisciplinary design optimization (MDO) that is able to treat both linking variables and linking functions (i.e. system-wide objectives and constraints). Contrary to quasi-separable problems with only linking variables, the presence of linking functions may hinder the parallel solution of subproblems and the use of the efficient alternating directions method of multipliers. We show that this unfortunate situation is not the case for MDO problems with block-separable linking constraints. We derive a centralized formulation of ALC for block-separable constraints, which does allow parallel solution of subproblems. Similarly, we derive a distributed coordination variant for which subproblems cannot be solved in parallel, but that still enables the use of the alternating direction method of multipliers. The approach can also be used for other existing MDO coordination strategies such that they can include block-separable linking constraints.

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>

Augmented Lagrangian coordination for distributed optimal design in MDO

AbstractQuite a number of coordination methods have been proposed for the distributed optimal design of large‐scale systems consisting of a number of interacting subsystems. Several coordination methods are known to have numerical convergence difficulties that can be explained theoretically. The methods for which convergence proofs are available have mostly been developed for so‐called quasi‐separable problems (i.e. problems with individual subsystems coupled only through a set of linking variables, not through constraints and/or objectives). In this paper, we present a new coordination approach for multidisciplinary design optimization problems with linking variables as well as coupling objectives and constraints. Two formulation variants are presented, offering a large degree of freedom in tailoring the coordination algorithm to the design problem at hand. The first, centralized variant introduces a master problem to coordinate coupling of the subsystems. The second, distributed variant coordinates coupling directly between subsystems. Our coordination approach employs an augmented Lagrangian penalty relaxation in combination with a block coordinate descent method. The proposed coordination algorithms can be shown to converge to Karush–Kuhn–Tucker points of the original problem by using the existing convergence results. We illustrate the flexibility of the proposed approach by showing that the analytical target cascading method of Kim et al. (J. Mech. Design‐ASME 2003; 125(3):475–480) and the augmented Lagrangian method for quasi‐separable problems of Tosserams et al. (Struct. Multidisciplinary Opt. 2007, to appear) are subclasses of the proposed formulations. Copyright © 2007 John Wiley &amp; Sons, Ltd.