Performance Evaluation of Augmented Lagrangian Coordination for Distributed Multidisciplinary Design Optimization

Abstract

The numerical performance and formulation flexibility of an augmented Lagrangian coordination method proposed by the authors is demonstrated on two example problems. First, a geometric programming problem is decomposed in a number of different ways to illustrate the flexibility of the approach in setting up different coordination structures. For this problem and a business jet design example, numerical results indicate that the coordination method is effective and robust in finding (local) solutions of the original non-decomposed problem, and does not introduce new local minima. The required coordination costs are found to be determined by how the problem is partitioned and coordinated. These costs do not only depend on the number of quantities that have to be coordinated, but also on their coupling strengths. The formulation flexibility of the method provides means to minimize these costs by adapting the decomposition to a problem at hand. The field of multidisciplinary design optimization (MDO) is concerned with the design of large-scale engineering systems that consist of a number of interacting subsystems. The size and the required level of expertise of each subsystem often prohibits the design of these large-scale systems to be performed in an integrated fashion. Instead, the problem is decomposed into smaller, more manageable parts, or design subproblems. To deal with the resulting coupled subproblems, a systematical coordination approach to system design is required. Many coordination methods have been proposed for the distributed optimal design of MDO problems. These coordination methods include Concurrent SubSpace Optimization (CSSO), 1 the Bi-Level Integrated System Synthesis method (BLISS/BLISS2000), 2,3 Collaborative Optimization (CO), 4,5 and the constraint margin approach of Haftka and Watson. 6 Several of these coordination methods may experience numerical difficulties when solving the master problem due to non-smoothness or failure to meet certain constraint qualifications. 7‐9 This may hinder the use of existing efficient gradient-based solution algorithms such as Sequential Quadratic Programming (SQP). Methods that do not satisfy these requirements have to use specialized, typically inefficient algorithms to solve the associated optimization problems. Refs. [3,10,11] propose the use of response surfaces to circumvent the difficulties due to the nonsmoothness. During the last years, several new penalty function-based coordination methods have been developed: Analytical Target Cascading (ATC), 12‐14 the penalty decomposition methods (IPD/EPD) of Ref. [15], and the augmented Lagrangian decomposition method (ALD) of Ref. [16]. For these methods, basic constraint qualifications hold, and the optimization (sub)problems are smooth. All methods can be shown to converge to the optimal solution of the original problem under certain assumptions such as smoothness and/or convexity. The formulation of IPD/EPD is nested, similar to CO, and for every function evaluation of the master problem, an optimization of the subproblems is necessary. ATC and ALD follow an alternating approach that iterates between solving the master problem and the subproblems. IPD/EPD and ALD coordinate “classic” MDO problems in a bi-level fashion by introducing a master problem that is superimposed over a number of subproblems, each associated with one of the subsystems. ATC can be applied to multi-level problems that may consist of more than two levels. All three methods require the problem to be quasiseparable: 6 Problems may only be coupled through shared variables; coupling objectives and constraints are not allowed.