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Divergent exploration in design with a dynamic multiobjective optimization formulation

Abstract

Formulation space exploration is a new strategy for multiobjective optimization that facilitates both divergent exploration and convergent optimization during the early stages of design. The formulation space is the union of all variable and design objective spaces identified by the designer as being valid and pragmatic problem formulations. By extending a computational search into the formulation space, the solution to an optimization problem is no longer predefined by any single problem formulation, as it is with traditional optimization methods. Instead, a designer is free to change, modify, and update design objectives, variables, and constraints and explore design alternatives without requiring a concrete understanding of the design problem a priori. To facilitate this process, we introduce a new vector/matrix-based definition for multiobjective optimization problems, which is dynamic in nature and easily modified. Additionally, we provide a set of exploration metrics to help guide designers while exploring the formulation space. Finally, we provide an example to illustrate the use of this new, dynamic approach to multiobjective optimization.

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Acknowledgement

This research was partially supported by National Science Foundation Grant 0954580.

Author information

Correspondence to C. A. Mattson.

Appendix: Pseudo Codes

Appendix: Pseudo Codes

Standard Multiobjective Optimization Code (P1)

  1. 1

    Define Design Variable Limits...

  2. 2

    Define Design Parameter Values...

  3. 3

    Construct x0

  4. 4

    Construct xL

  5. 5

    Construct xU

  6. 6

    Construct P

  7. 7

    Call [x*, mu*] = optimize(x0, xL, xU, P)

  8. 8

     

  9. 9

    function [objectives] = objectiveFunction(x, P)

  10. 10

    Call [outputs] = model(x, P)

  11. 11

    Calculate objectives

  12. 12

     

  13. 13

    function [g, h] = constraintFunction(x, P)

  14. 14

    Call [outputs] = model(x, P)

  15. 15

    Define Equality Constraint Values...

  16. 16

    Calculate h...

  17. 17

    Define Inequality Constraint Values...

  18. 18

    Calculate g...

  19. 19

     

  20. 20

    function [outputs] = model(x, P)

  21. 21

    Extract x...

  22. 22

    Extract P...

  23. 23

    Calculate outputs...

Dynamic Multiobjective Optimization Code (P2)

  1. 1

    Define Independent Design Object Limits...

  2. 2

    Define Dependent Design Object Limits...

  3. 3

    Construct y0

  4. 4

    Construct yL

  5. 5

    Construct yU

  6. 6

    Construct zL

  7. 7

    Construct zU

  8. 8

    Define w...

  9. 9

    Call [x*] = optimize(y0, yL, yU, zL, zU, w)

  10. 10

     

  11. 11

    function [objectives] = objectiveFunction(y, w)

  12. 12

    Call [z] = model(y)

  13. 13

    Calculate x = [y;z]

  14. 14

    Calculate objectives = w*x

  15. 15

     

  16. 16

    function [constraints] = constraintFunction(y, zL, zU)

  17. 17

    Call [z] = model(y)

  18. 18

    Calculate constraints = [zL-z; z-zU]

  19. 19

     

  20. 20

    function [z] = model(y)

  21. 21

    Extract y...

  22. 22

    Calculate z...

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Curtis, S.K., Mattson, C.A., Hancock, B.J. et al. Divergent exploration in design with a dynamic multiobjective optimization formulation. Struct Multidisc Optim 47, 645–657 (2013). https://doi.org/10.1007/s00158-012-0855-8

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Keywords

  • Conceptual design
  • Multiobjective optimization
  • Design space exploration