Distributed optimization for systems design : an augmented Lagrangian coordination method

This thesis presents a coordination method for the distributed design optimization of engineering systems. The design of advanced engineering systems such as aircrafts, automated distribution centers, and microelectromechanical systems (MEMS) involves multiple components that together realize the desired functionality. The design requires detailed knowledge of the various physics that play a role in each component. Since a single designer is no longer able to oversee all relevant aspects of the complete system, the design process is distributed over a number of design teams. Each team autonomously operates on a single component or aspect of the system, and uses dedicated analysis and design tools to solve the specific problems encountered for this subsystem. Consequently, one team is often not familiar with the design considerations of other teams, and does not know how its decisions affect the system as a whole. An additional challenge in systems design is introduced by market competitiveness and increasing consumer requirements, which pushes systems towards the limits of performance and cost. Since each subsystem contributes to the system performance, the interaction between these subsystems, and thus design teams, becomes critical and needs to be controlled. Design optimization is an effective and powerful approach to finding optimal designs when parametric models are available to describe the relevant system behavior. To fully exploit the available design expertise, a coordination approach for distributed system optimization is required that respects the organization of design teams and their tools. The augmented Lagrangian coordination (ALC) method presented in this thesis is a coordination approach for distributed optimization that a) provides disciplinary design autonomy, b) offers the designer a large degree flexibility in setting up the coordination structure, c) maintains mathematical rigor, and d) is efficient in obtaining optimal and consistent designs. ALC is based on a combination of augmented Lagrangian relaxation and block-coordinate descent, two techniques from mathematical programming. What distinguishes ALC from other coordination methods for distributed system design is its flexibility in combination with its mathematical rigor. The flexibility relates both to the structure of the coordination process, and to the type of coupling that is allowed. Most coordination methods with convergence proof follow a hierarchical structure in which a single master problem is in charge of coordination. The master problem is superimposed over the disciplinary subproblems that communicate only with this master problem. ALC allows a more general, non-hierarchical coordination structure in which coordination may also be performed between disciplinary subproblems directly. Furthermore, ALC can coordinate not only linking variables, but also coupling functions. The mathematical rigor assures that, under suitable assumptions, the solutions obtained with ALC algorithms are optimal and consistent. Specialized ALC algorithms based on the efficient alternating direction method of multipliers are developed for problems that have only linking variables and/or block-separable coupling constraints. Furthermore, we demonstrate that the well-known analytical target cascading method is a subclass of ALC. ALC algorithms can be proven to converge to locally optimal and consistent designs under smoothness assumptions, provided that subproblem solutions are globally optimal. Global optimality is however difficult, if not impossible, to guarantee in practice since many engineering design problems are non-convex. When only local optimality can be obtained, ALC methods can no longer be proven to yield optimal or consistent solutions. Experiments with several non-convex problems show however that ALC with locally optimal solutions to subproblems often still converges to a local or global system optimum; however, occasionally inconsistent designs are encountered. Performing a global search at subproblems improves the convergence behavior, and globally optimal solutions are frequently obtained. The work in this thesis is part of MicroNed, a national research programme on microsystem technology. In the emerging field of microsystem technology, optimization of cost, size, and performance are very relevant since these factors determine whether microsystems can be successful alternatives for existing "macro" systems. Proper functioning of the microdevice may increasingly depend on model-based optimization during the design. To illustrate how coordination methods can be used in microsystem optimal design, a micro-accelerometer design problem has been developed, inspired on a commercially available device. The goal of the design problem is to find the dimensions of the accelerometer such that its area is minimized subject to performance requirements on sensitivity, noise, and bandwidth, while considering mechanical, electrostatic, dynamic, and electrical constraints. The behavioral models are analytical, providing a reproducible benchmark problem for performance assessments of coordination methods in distributed optimization.