Analytical target cascading is a method for design optimization of hierarchical, multilevel systems. A quadratic penalty relaxation of the system consistency constraints is used to ensure subproblem feasibility. A typical nested solution strategy consists of inner and outer loops. In the inner loop, the coupled subproblems are solved iteratively with fixed penalty weights. After convergence of the inner loop, the outer loop updates the penalty weights. The article presents an augmented Lagrangian relaxation that reduces the computational cost associated with ill-conditioning of subproblems in the inner loop. The alternating direction method of multipliers is used to update penalty parameters after a single inner loop iteration, so that subproblems need to be solved only once. Experiments with four examples show that computational costs are decreased by orders of magnitude ranging between 10 and 1000.

SUMMARYQuite a number of coordination methods have been proposed for the distributed optimal design of largescale 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.

Several decomposition methods have been proposed for the distributed optimal design of quasi-separable problems encountered in Multidisciplinary Design Optimization (MDO). Some of these methods are known to have numerical convergence difficulties that can be explained theoretically. We propose a new decomposition algorithm for quasi-separable MDO problems. In particular, we propose a decomposed problem formulation based on the augmented Lagrangian penalty function and the block coordinate descent algorithm. The proposed solution algorithm consists of inner and outer loops. In the outer loop, the augmented Lagrangian penalty parameters are updated. In the inner loop, our method alternates between solving an optimization master problem, and solving disciplinary optimization subproblems. The coordinating master problem can be solved analytically; the disciplinary subproblems can be solved using commonly available gradient-based optimization algorithms. The augmented Lagrangian decomposition method is derived such that existing proofs can be used to show convergence of the decomposition algorithm to KKT points of the original problem under mild assumptions. We investigate the numerical performance of the proposed method on two example problems.

This paper presents a classification of formulations for distributed system optimization based on formulation structure. Two main classes are identified: nested formulations and alternating formulations. Nested formulations are bilevel programming problems where optimization subproblems are nested in the functions of a coordinating master problem. Alternating formulations iterate between solving a master problem and disciplinary subproblems in a sequential scheme. Methods included in the former class are collaborative optimization and BLISS2000. The latter class includes concurrent subspace optimization, analytical target cascading, and augmented Lagrangian coordination. Although the distinction between nested and alternating formulations has not been made in earlier comparisons, it plays a crucial role in the theoretical and computational properties of distributed optimization methods. The most prominent general characteristics for each class are discussed in more detail, providing valuable insights for the theoretical analysis and further development of distributed optimization methods.

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.