An Augmented Lagrangian Relaxation for Analytical Target Cascading using the Alternating Directions Method of Multipliers

Analytical Target Cascading (ATC) is a method for design optimization of hierarchically decomposed multilevel systems. ATC subproblems are defined by introducing target and response variables that couple the subsystems of the original system. During the iterative solution inconsistencies between target and response variable values are minimized using a quadratic penalty function. Typically, a nested solution strategy is used consisting of an inner and an outer loop. In the inner loop subproblems are solved with fixed penalty weights while in the outer loop these weights are updated with informationfrom the inner loop. Two sources of computational cost associated with solving the decomposed ATCproblem are observed. First, accurate solutions can often be obtained only with large penalty weights,which can also introduce ill-conditioning of the subproblems. Second, subproblems are not independent and their solution has to be coordinated within the inner loop, meaning that subproblems may have to be solved many times before the algorithm can return to the outer loop. The article introduces the use of an augmented Lagrangian function to obtain accurate subproblem solutions for relatively small weights.To reduce the computational cost of coordination in the inner loop, an alternating directions method of multipliers is used. Instead of updating penalty parameters at convergence of the inner loop, the alternating direction method updates the penalty parameters after a single inner loop iteration. Innerloop coordination is reduced to solving subproblems only once. These new strategies are demonstrated on two example problems and compared to the quadratic penalty function currently used for ATC.Computational costs for the tested problems are decreased by orders of magnitude ranging between ten and one thousand.