Dual Residual for Distributed Augmented Lagrangian Coordination Based on Optimality Conditions

Augmented Lagrangian Coordination (ALC) is one of the more popular coordination strategies for decomposition based optimization. It employs the augmented Lagrangian relaxation approach and has shown great improvements in terms of efficiency and solution accuracy when compared to other methods addressing the same type of problem. Additionally, by offering two variants: the centralized ALC in which an artificial master problem in the upper level is created to coordinate all the sub-problems in the lower level, and the distributed ALC in which coordination can be performed directly between sub-problems without a master problem, ALC provides more flexibility than other methods. However, the initial setting and the update strategy of the penalty weights in ALC still significantly affect its performance and thus are worth further research. For centralized ALC, the non-monotone weight update strategy based on the theory of dual residual has shown very good improvements over the traditional monotone update, in which the penalty weights can either increase or decrease. In this paper, we extend the research on the dual residual in centralized ALC to the distributed ALC. Through applying the Karush-Kuhn-Tucker (KKT) optimality conditions to the All-In-One (AIO) and decomposed problems, the necessary conditions for the decomposed solution to be optimal are derived, which leads to the definition of primal and dual residuals in distributed ALC. A new non-monotone weight based on both residuals is then proposed, by which all AIO KKT conditions are guaranteed after decomposition. Numerical tests are conducted on two mathematical problems and one engineering problem and the performances of the new update are compared to those of the traditional update. The results show that our proposed methods improve the process efficiency, accuracy, and robustness for distributed ALC.

Dual Residual in Augmented Lagrangian Coordination for Decomposition-Based Optimization

As system design problems increase in complexity, researchers seek approaches to optimize such problems by coordinating the optimizations of decomposed sub-problems. Many methods for optimization by decomposition have been proposed in the literature among which, the Augmented Lagrangian Coordination (ALC) method has drawn much attention due to its efficiency and flexibility. The ALC method involves a quadratic penalty term, and the initial setting and update strategy of the penalty weight are critical to the performance of the ALC. The weight in the traditional weight update strategy always increases and previous research shows that an inappropriate initial value of the penalty weight may cause the method not to converge to optimal solutions. Inspired by the research on Augmented Lagrangian Relaxation in the convex optimization area, a new weight update strategy in which the weight can either increase or decrease is introduced into engineering optimization. The derivation of the primal and dual residuals for optimization by decomposition is conducted as a first step. It shows that the traditional weight update strategy only considers the primal residual, which may result in a duality gap and cause a relatively big solution error. A new weight update strategy considering both the primal and dual residuals is developed which drives the dual residual to zero in the optimization process, thus guaranteeing the solution accuracy of the decomposed problem. Finally, the developed strategy is applied to both mathematical and engineering test problems and the results show significant improvements in solution accuracy. Additionally, the proposed approach makes the ALC method more robust since it allows the coordination to converge with an initial weight selected from a much wider range of possible values while the selection of initial weight is a big concern in the traditional weight update strategy.

MULTI-OPTIONS BIDDING STRATEGY IN DISTRIBUTED ENVIRONMENT

When a Make-To-Order (MTO) enterprise receives a bidding invitation from a customer, he does not know the demand is price-sensitive or time-sensitive. In order to increase the possibility of winning this contingent order, the enterprise tenders his bid with two options characterized with price and delivery time. One is for time-sensitive demand, and it ensures that the products will be delivered as soon as possible. The other is for price-sensitive demand, and it ensures that the products will be produced at the lowest cost. The enterprise is comprised of several plants that jointly produce custom products and each plant's decision is made in a distributed way. Hence, the enterprise has to coordinate the production plans of these plants to generate the two options. Since Analytical Target Cascading (ATC) can solve multi-level hierarchical distributed problem and its convergence is proven, it is adopted to coordinate the plants' planning. This paper proposes a unique bidding option generating process in which an extended ATC model is utilized to produce time-sensitive and price-sensitive options by adjusting the coefficient of delivery tardiness. The numerical study in this paper demonstrates (1) the extended ATC model works effectively; (2) ATC with augmented Lagrangian relaxation (AL-ATC) is more effective in producing solutions than ATC with quadratic penalty function (QP-ATC); (3) ATC is an effective coordination tool by comparing the solutions made by ATC and All-In-One (AIO).