In real life applications, optimization problems with more than one objective function are often of interest. Next to handling multiple objective functions, another challenge is to deal with uncertainties concerning the realization of the decision variables. One approach to handle these uncertainties is to consider the objectives as set-valued functions. Hence, the image of one decision variable is a whole set, which includes all possible outcomes of this decision variable. We choose a robust approach and thus these sets have to be compared using the so-called upper-type less order relation. We propose a numerical method to calculate a covering of the set of optimal solutions of such an uncertain multiobjective optimization problem. We use a branch-and-bound approach and lower and upper bound sets for being able to compare the arising sets. The calculation of these lower and upper bound sets uses techniques known from global optimization, as convex underestimators, as well as techniques used in convex multiobjective optimization as outer approximation techniques. We also give first numerical results for this algorithm.
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The second author thanks the Carl-Zeiss-Stiftung and the DFG-founded Research Training Group 1567 “Lorentz Force Velocimetry and Lorentz Force Eddy Current Testing” for financial support. The work of the third author is funded by the Deutsche Forschungsgemeinschaft under Grant No. EI 821/4. Additionally, we thank the two anonymous referees for their helpful remarks and comments, which helped us to improve this article.
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Eichfelder, G., Niebling, J. & Rocktäschel, S. An algorithmic approach to multiobjective optimization with decision uncertainty. J Glob Optim 77, 3–25 (2020). https://doi.org/10.1007/s10898-019-00815-9
- Multiobjective optimization
- Decision uncertainty
- Branch-and-bound algorithm