Convergence Strategy for Parallel Solving of Analytical Target Cascading with Augmented Lagrangian Coordination

Abstract

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 relaxation with Alternating Direction method (AL-AD) has been widely used for the coordination process of both hierarchical ATC 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 ATC is that it can solve subsystem problems in parallel, thus allowing it to reduce computational cost by parallel computing. Previous studies have proposed AL coordination strategies for parallelization by eliminating interactions among subproblems. This is done by introducing a master problem and support variables or by approximating a quadratic penalty term to make subproblems separable. However, conventional AL-AD does not guarantee convergence in the case of parallel solving. Our study found that, in parallel solving using targets and responses of the current iteration, conventional AL-AD causes mismatch of information in updating the Lagrange multiplier (LM). Therefore, the LM may not reach the optimal point, and as a result, increasing penalty weight causes numerical difficulty in the AL penalty function approach. To solve this problem, we propose a modified AL-AD for parallel solving in non-hierarchical ATC. The proposed algorithm uses the subgradient method with adaptive step size in updating the LM, which is independent of quadratic penalty terms and keeps quadratic penalty terms at the initial value. Without approximation or introduction of an artificial master problem, the modified AL-AD for parallel solving can achieve similar accuracy and convergence with much less computational cost, compared with conventional methods.