Abstract
In inverse optimization problems, the goal is to modify the costs in an underlying optimization problem in such a way that a given solution becomes optimal, while the difference between the new and the original cost functions, called the deviation vector, is minimized with respect to some objective function. The \(\ell _1\)- and \(\ell _{\infty }\)-norms are standard objectives used to measure the size of the deviation. Minimizing the \(\ell _1\)-norm is a natural way of keeping the total change of the cost function low, while the \(\ell _{\infty }\)-norm achieves the same goal coordinate-wise. Nevertheless, none of these objectives is suitable to provide a balanced or fair change of the costs.
In this paper, we initiate the study of a new objective that measures the difference between the largest and the smallest weighted coordinates of the deviation vector, called the weighted span. We provide a Newton-type algorithm for finding one that runs in strongly polynomial time in the case of unit weights.
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Notes
- 1.
The notion of span appears under several names in various branches of mathematics, such as range in statistics, amplitude in calculus, or deviation in engineering.
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Acknowledgement
This research has been implemented with the support provided by the Lendület Programme of the Hungarian Academy of Sciences – grant number LP2021-1/2021, by the Ministry of Innovation and Technology of Hungary – grant number ELTE TKP 2021-NKTA-62, and by Dynasnet European Research Council Synergy project – grant number ERC-2018-SYG 810115. The authors have no competing interests to declare that are relevant to the content of this article.
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Bérczi, K., Mendoza-Cadena, L.M., Varga, K. (2024). Newton-Type Algorithms for Inverse Optimization: Weighted Span Objective. In: Soto, J.A., Wiese, A. (eds) LATIN 2024: Theoretical Informatics. LATIN 2024. Lecture Notes in Computer Science, vol 14579. Springer, Cham. https://doi.org/10.1007/978-3-031-55601-2_22
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