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.