Optimization and Engineering

, Volume 19, Issue 4, pp 887–915 | Cite as

Relaxing high-dimensional constraints in the direct solution space method for early phase development

  • Volker A. LangeEmail author
  • Johannes Fender
  • Fabian Duddeck


Early phase distributed system design can be accomplished using solution spaces that provide an interval of permissible values for each functional parameter. The feasibility property guarantees fulfillment of all design requirements for all possible realizations. Flexibility denotes the size measure of the intervals, with higher flexibility benefiting the design process. Two methods are available for solution space identification. The direct method solves a computationally cheap optimization problem. The indirect method employs a sampling approach that requires a relaxation of the feasibility property through re-formulation as a chance constraint. Even for high probabilities of fulfillment, \(P>0.99\), this results in substantial increases in flexibility, which offsets the risk of infeasibility. This work implements the chance constraint formulation into the direct method for linear constraints by showing that its problem statement can be understood as a linear robust optimization problem. Approximations of chance constraints from the literature are transferred into the context of solution spaces. From this, we derive a theoretical value for the safety parameter \(\varOmega\). A further modification is presented for use cases, where some intervals are already predetermined. A problem from vehicle safety is used to compare the modified direct and indirect methods and discuss suitable choices of \(\varOmega\). We find that the modified direct method is able to identify solution spaces with similar flexibility, while maintaining its cost advantage.


Solution spaces Chance constraints Robust optimization High-dimensional geometry Vehicle design 



The authors would like to thank the three anonymous referees for valuable comments and express their sincere appreciation to Dr. Thomas G. Amler for the many fruitful discussions on Solution Spaces.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.BMW Group, Research and Innovation CenterMunichGermany
  2. 2.Technical University of MunichMunichGermany
  3. 3.Queen Mary University of LondonLondonUK

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