Abstract
Minimizing a convex function over a convex set is a basic algorithmic problem. We give a simple algorithm for the general setting in which the function is given by an evaluation oracle and the set by a membership oracle. The algorithm takes \(\widetilde{O}(n^{2})\) oracle calls and \(\widetilde{O}(n^{3})\) additional arithmetic operations. This results in more efficient reductions among the five basic oracles for convex sets and functions defined by Grötschel, Lovász and Schrijver (Algorithms Comb 2, (1988), [5]).
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Acknowledgements
We thank Sebastien Bubeck, Ben Cousins, Sham Kakade and Ravi Kannan for helpful discussions, Yan Kit Chim for making the illustrations, and Xiaodi Wu for pointing out some typos in a previous version of the paper. This work was supported in part by NSF Awards CCF-1563838, CCF-1717349 and CCF1740551.
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© 2019 János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature
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Lee, Y.T., Sidford, A., Vempala, S.S. (2019). Efficient Convex Optimization with Oracles. In: Bárány, I., Katona, G., Sali, A. (eds) Building Bridges II. Bolyai Society Mathematical Studies, vol 28. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59204-5_10
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