Minimization of Even Conic Functions on the Two-Dimensional Integral Lattice

Abstract

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.