Abstract
We consider the minimum-norm-point (MNP) problem of polyhedra, a well-studied problem that encompasses linear programming. Inspired by Wolfe’s classical MNP algorithm, we present a general algorithmic framework that performs first order update steps, combined with iterations that aim to ‘stabilize’ the current iterate with additional projections, i.e., finding a locally optimal solution whilst keeping the current tight inequalities. We bound the number of iterations polynomially in the dimension and in the associated circuit imbalance measure. In particular, the algorithm is strongly polynomial for network flow instances. The conic version of Wolfe’s algorithm is a special instantiation of our framework; as a consequence, we obtain convergence bounds for this algorithm. Our preliminary computational experiments show a significant improvement over standard first-order methods.
This is an extended abstract. The full version including all omitted proofs is available on arXiv:2211.02560.
SF’s research is supported by JSPS KAKENHI Grant Numbers JP19K11839 and 22K11922 and by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. TK is supported by JSPS KAKENHI Grant Number JP19K11830. LAV’s research is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 757481–ScaleOpt).
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Acknowledgments
The third author would like to thank Richard Cole, Daniel Dadush, Christoph Hertrich, Bento Natura, and Yixin Tao for discussions on first order methods and circuit imbalances.
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Fujishige, S., Kitahara, T., Végh, L.A. (2023). An Update-and-Stabilize Framework for the Minimum-Norm-Point Problem. In: Del Pia, A., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2023. Lecture Notes in Computer Science, vol 13904. Springer, Cham. https://doi.org/10.1007/978-3-031-32726-1_11
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