Given a separation oracle $\mathsf{SO}$ for a convex function $f$ that has an integral minimizer inside a box with radius $R$, we show how to efficiently find a minimizer of $f$ using at most $O(n (n + \log(R)))$ calls to $\mathsf{SO}$. When the set of minimizers of $f$ has integral extreme points, our algorithm outputs an integral minimizer of $f$. This improves upon the previously best oracle complexity of $O(n^2 (n + \log(R)))$ obtained by an elegant application of simultaneous diophantine approximation due to [Gr\"otschel, Lov\'asz and Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. We conjecture that our oracle complexity is tight up to constant factors for polynomial time algorithms. Our result immediately implies a strongly polynomial algorithm for the Submodular Function Minimization problem that makes at most $O(n^3)$ calls to an evaluation oracle. This improves upon the previously best $O(n^3 \log^2(n))$ oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, V\'egh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity $O(n^3 \log(n))$ given in the former work. Our result is achieved by an application of the LLL algorithm [Lenstra, Lenstra and Lov\'asz, Math. Ann. 1982] for the shortest lattice vector problem. We show how an approximately shortest vector of certain lattice can be used to reduce the dimension of the problem, and how the oracle complexity of such a procedure is advantageous compared with the Gr\"otschel-Lov\'asz-Schrijver approach that uses simultaneous diophantine approximation. Our analysis of the oracle complexity is based on a potential function that captures simultaneously the size of the search set and the density of the lattice. To achieve the $O(n^2)$ term in the oracle complexity, technical ingredients from convex geometry are applied.

中文翻译：

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使用积分最小化器最小化凸函数

给定一个分离 oracle $\mathsf{SO}$ 的凸函数 $f$，它在半径为 $R$ 的盒子内有一个积分极小值，我们展示了如何有效地找到 $f$ 的极小值使用至多 $O( n (n + \log(R)))$ 调用 $\mathsf{SO}$。当 $f$ 的极小值集合具有积分极值点时，我们的算法输出 $f$ 的积分极小值。由于 [Gr\"otschel, Lov\'asz 和 Schrijver, Prog. Comb. Opt. 1984, Springer 1988] 三十多年前。我们推测，我们的预言机复杂性非常接近多项式时间算法的常数因子。我们的结果立即暗示了用于子模函数最小化问题的强多项式算法，该算法最多对评估预言机进行 $O(n^3)$ 调用。这改进了 [Lee, Sidford and Wong, FOCS 2015] 和 [Dadush, V\'egh and Zambelli, SODA 2018]，以及之前工作中给出的具有预言机复杂度 $O(n^3 \log(n))$ 的指数时间算法。我们的结果是通过应用 LLL 算法实现的 [Lenstra, Lenstra and Lov\'asz, Math. 安。1982] 最短格向量问题。我们展示了如何使用某个格的近似最短向量来降低问题的维数，以及与 Gr\"otschel-Lov\' 相比，这种过程的预言复杂性如何具有优势 asz-Schrijver 方法使用同步丢番图近似。我们对预言机复杂性的分析基于一个潜在函数，该函数同时捕获搜索集的大小和格子的密度。为了在 oracle 复杂性中实现 $O(n^2)$ 项，应用了凸几何的技术成分。