### Article

## Minimization of even conic functions on the two-dimensional integral lattice

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*log_2(R)+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*log_2(R) + O(1).We give an 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_2(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.