Computer Science/Discrete Mathematics Seminar I | |

Topic: | Title : An Improved Cutting Plane Method for Convex Optimization, Convex-Concave Games and its Applications |

Speaker: | Zhao Song |

Affiliation: | Member, School of Mathematics |

Date: | Monday, January 27 |

Time/Room: | 11:00am - 12:00pm/Simonyi Hall 101 |

â€‹Given a separation oracle for a convex set $K \subset \mathbb{R}^n$ that is contained in a box of radius $R$, the goal is to either compute a point in $K$ or prove that $K$ does not contain a ball of radius $\epsilon$. We propose a new cutting plane algorithm that uses an optimal $O(n \log (\kappa))$ evaluations of the oracle and an additional $O(n^2)$ time per evaluation, where $\kappa = nR/\epsilon$.

- This improves upon Vaidya's $O( \text{SO} \cdot n \log (\kappa) + n^{\omega+1} \log (\kappa))$ time algorithm [Vaidya, FOCS 1989a] in terms of polynomial dependence on $n$, where $\omega < 2.373$ is the exponent of matrix multiplication and $\text{SO}$ is the time for oracle evaluation.
- This improves upon Lee-Sidford-Wong's $O( \text{SO} \cdot n \log (\kappa) + n^3 \log^{O(1)} (\kappa))$ time algorithm [Lee, Sidford and Wong, FOCS 2015] in terms of dependence on $\kappa$.