Pareto set analysis: local measures of objective coupling in multiobjective design optimization

• Published in: Structural and multidisciplinary optimization Vol. 43; no. 5; pp. 617 - 630
• Main Authors: Frischknecht, Bart D., Peters, Diane L., Papalambros, Panos Y.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.05.2011; Springer Nature B.V
• Subjects: Alignment; Co-design; Computational Mathematics and Numerical Analysis; Control systems design; Coupling; Design optimization; Engineering; Engineering Design; Mathematical analysis; Multiple objective analysis; Objectives; Optimal control; Pareto optimization; Pareto optimum; Research Paper; Theoretical and Applied Mechanics; Co-design; Pareto set; Sensitivity analysis; Objective coupling; Multiobjective optimization; Design optimization
• ISSN: 1615-147X; 1615-1488
• DOI: 10.1007/s00158-010-0599-2

Multiobjective optimization focuses on the explicit trade-offs between competing criteria. A particular case is the study of combined optimal design and optimal control, or co-design, of smart artifacts where the artifact design and controller design objectives compete. In the system-level co-design problem, the objective is often the weighted sum of these two objectives. A frequently referenced practice is to solve co-design problems in a sequential manner: design first, control next. The success of this approach depends on the form of coupling between the two subproblems. In this paper, the coupling vector derived for a system problem with unidirectional coupling is shown to be related to the alignment of competing objectives, as measured by the polar cone of objective gradients, in the bi-objective programming formulation. Further, it is shown that a measure describing the case where a range of objective weighting values for the system objective result in identical design solutions can be normalized when the system problem is considered as a bi-objective one. Changes to the mathematical structure and input parameter values of a bi-objective programming problem can lead to changes in the shape of the attainable set and its Pareto boundary. We illustrate the link between the coupling and alignment measures and the outcomes of the Pareto set. Systematically studying changes to coupling and alignment measures due to changes to the multiobjective formulation can yield deeper insights into the system-level design problem. Two examples illustrate these results.