Under consideration is the Successive Minima Problem for the 2-dimensional lattice with respect to the order given by some conic function f. We propose an algorithm with complexity of 3.32 log₂R + O(1) calls to the comparison oracle of f, where R is the radius of the circular searching area, while the best known lower oracle complexity bound is 3 log₂R + O(1). Wegivean efficient criterion for checking that given vectors of a 2-dimensional lattice are successive minima and form a basis for the lattice. Moreover, we show that the similar Successive Minima Problem for dimension n can be solved by an algorithm with at most O(n)²ⁿ log R calls to the comparison oracle. The results of the article can be applied to searching successive minima with respect to arbitrary convex functions defined by the comparison oracle.

中文翻译：

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二维积分格上偶圆锥函数的最小化

正在考虑的是关于二次曲线函数f给出的2维晶格的连续最小问题。我们提出了一种算法，其复杂度为3.32 log ₂ R + O（1）调用f的比较oracle ，其中R是圆形搜索区域的半径，而最著名的下级oracle复杂度下界为3 log ₂ R + O（1）。Wegivean有效准则，用于检查二维晶格的给定矢量是连续的最小值，并构成该晶格的基础。此外，我们表明，对于维数为n的相似的连续极小问题，可以用最多O（n）^{2 n} log R调用比较oracle。本文的结果可用于搜索由比较Oracle定义的任意凸函数的连续最小值。