Within the classical design of, for example, bridges and airplane structures, optimization approaches generally focus primarily on adjusting large-scale (macroscopic) parameters and neglect the influence of small-scale (microscopic) parameters on the large-scale behavior. The main reason is because of numerical efficiency. Multi-level optimization techniques rely on a decomposition of the optimization problem into separate levels or subsystems. Thus, it is attempted to incorporate design variables originating from different (multiple) levels in a cost-effective manner. In literature various multi-level optimization approaches are described in which six main approaches can be distinguished, Optimization by Linear Decomposition, Collaborative Optimization, Concurrent SubSpace Optimization, Bi-Level Integrated System Synthesis, Analytical Target Cascading and the method of Quasi-separable Subsystem Decomposition. However, a comparative overview of these methods is still lacking. In this study these multi-level optimization schemes are compared on the basis of treatment of interdisciplinary consistency constraints between the hierarchical levels.
Multilevel optimization methods use the capability of an optimization problem to be decomposed into separate levels or subsystems. In order to decompose the coupled analysis, linking variables have to be introduced. A clear distinction between the aspects of optimization and analysis in multi-level design is made using a new multi-level notation. It emphasizes the handling of inconsistencies between subsystems and their solution. The proposed notation enables a clear comparison of the multi-level methods on the basis of their treatment of interdisciplinary consistency constraints and allows classification on the basis of handling of constraints, as previously demonstrated by Alexandrov and Lewis [
] for two multi-level optimization formulations. In this paper, this classification of multi-level optimization approaches is extended to include all six multi-level formulations.
Furthermore, the multi-level optimization framework is demonstrated by applying the six multi-level methods to a classical two bar truss example. The work here is part of a larger ongoing effort towards integrating multi-scale mechanics and multi-level optimization. The two bar truss example is in this context a suited example. It contains relevant characteristics of multi-scale mechanics and multi-level optimization and its simplicity helps to capture the essence of multi-level design approaches. In future work we aim to describe multi-scale mechanics in a similar fashion, which is expected to open up possibilities of exploiting synergy effects through further integration of the two formulations.