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Inverse conic linear programs in Banach spaces

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Abstract

Given the costs and a feasible solution for a finite-dimensional linear program (LP), inverse optimization involves finding new costs that are close to the original and that also render the given solution optimal. Ahuja and Orlin employed the absolute sum norm and the maximum absolute norm to quantify distances between cost vectors, and applied duality to establish that the inverse LP problem can be formulated as another finite-dimensional LP. This was recently extended to semi-infinite LPs, countably infinite LPs, and finite-dimensional conic optimization problems. These works provide sufficient conditions so that the inverse problem also belongs to the same class as the forward problem. This paper extends this result to conic LPs in potentially infinite-dimensional Banach spaces. Moreover, the paper presents concrete derivations for continuous conic LPs, whose special cases include continuous linear programs and continuous conic programs; normed cone programs in Banach spaces, which include second-order cone programs as a special case; and semi-definite programs in Hilbert spaces. These derivations reveal the sharper result that, in each case, the inverse problem belongs to the same specific subclass as the forward problem. Instances where existing forward algorithms can then be adapted to solve the inverse problems are identified. Results in this paper may enable the application of inverse optimization to as yet unexplored areas such as continuous-time economic planning, continuous-time job-shop scheduling, continuous-time network flow, maximum flow with time-varying edge-capacities, and wireless optimization with time-varying coverage requirements.

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Acknowledgements

This research was funded in part by the National Science Foundation [grant #CMMI 1561918]. The author is grateful to an anonymous reviewer of a previous version of this paper, whose suggestions shortened and improved the presentation; helped merge two longer sections into a single Sect. 4; strengthened the proof of Lemma 2 by including measure-theoretic issues that the author had overlooked; and simplified the proof of Lemma 6 via a concrete approach instead of relying on a more general result from [32].

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Ghate, A. Inverse conic linear programs in Banach spaces. Optim Lett 15, 289–310 (2021). https://doi.org/10.1007/s11590-020-01683-z

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