Abstract
This paper studies a bilevel polynomial program involving box data uncertainties in both its linear constraint set and its lower-level optimization problem. We show that the robust global optimal value of the uncertain bilevel polynomial program is the limit of a sequence of values of Lasserre-type hierarchy of semidefinite linear programming relaxations. This is done by first transforming the uncertain bilevel polynomial program into a single-level non-convex polynomial program using a dual characterization of the solution of the lower-level program and then employing the powerful Putinar’s Positivstellensatz of semi-algebraic geometry. We provide a numerical example to show how the robust global optimal value of the uncertain bilevel polynomial program can be calculated by solving a semidefinite programming problem using the MATLAB toolbox YALMIP.
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Acknowledgements
The authors would like to thank the referees for the valuable comments and suggestions, which have improved the final preparation of the paper. The authors are grateful to Dr Guoyin Li for discussing about the Slater condition and for his help in the computer implementation of our methods. Research of the first author was supported by the UNSW Vice-Chancellor’s Postdoctoral Research Fellowship (RG134608/SIR50). Research of the second author was partially supported by a grant from the Australian Research Council.
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Chuong, T.D., Jeyakumar, V. Finding Robust Global Optimal Values of Bilevel Polynomial Programs with Uncertain Linear Constraints. J Optim Theory Appl 173, 683–703 (2017). https://doi.org/10.1007/s10957-017-1069-4
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DOI: https://doi.org/10.1007/s10957-017-1069-4
Keywords
- Bilevel programming
- Robust optimization
- Uncertain linear constraints
- Global polynomial optimization
- Semidefinite program