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Tropical Optimization


   School of Mathematics


About the Project

Tropical algebra is linear algebra developed over the max-plus semiring (extended real line endowed with tropical addition `a+b'=max(a,b) and tropical multiplication `ab'=a+b). In the project you will work on some optimisation problems expressed in terms of tropical algebra (in other words, optimisation problems over max-plus semiring). One of the three research directions will be pursued, based on your research preference: 1) tropical optimisation using mean-payoff game solvers, 2) bi-level tropical optimisation, 3) optimisation over symmetrised tropical semiring. Finding practical applications for optimisation problems being formulated and solved will be also a priority. See the suggested references for more background and existing results.


References

1. F.L. Baccelli et al. Synchronization and Linearity. Wiley, 1992. Available online: https://www.rocq.inria.fr/metalau/cohen/documents/BCOQ-book.pdf
2. P. Butkovic. Max-linear Systems: Theory and Applications. Springer, 2010.
3. S. Gaubert et al. Tropical linear-fractional programming and parametric mean-payoff games. Journal of Symbolic Computation 47(12), 2012, 1447-1478. Link: https://doi.org/10.1016/j.jsc.2011.12.049. Arxiv preprint: https://arxiv.org/abs/1101.3431
4. J. Parsons et al. Tropical psedoquatratic optimization as parametric mean-payoff games. Optimization, published online, 2022.
Link: https://doi.org/10.1080/02331934.2022.2085100 Arxiv preprint: https://arxiv.org/abs/2009.129305.
5. S. Sergeev and Z. Liu. Tropical analogues of a Dempe-Franke bilevel optimization problem,
In H.A. Le Thi et al. (Eds.) "Optimization of Complex Systems: Theory, Models, Algorithms and Applications", Springer, Cham, 2020, pp. 691-701. Arxiv preprint: https://arxiv.org/abs/1902.10055
6. N. Krivulin Direct solution to constrained tropical optimization problems with application to project scheduling. Comput. Management Sci. 14(1), 2017, pp. 91–113. Arxiv preprint: https://arxiv.org/abs/1501.075917.
7. X. Allamigeon et al. Tropicalizing the simplex algorithm. SIAM Journal on Discrete Mathematics 29(2), 2015, pp. 751-795. Link: https://doi.org/10.1137/130936464. Arxiv preprint: https://arxiv.org/abs/1308.0454

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