A polynomial algorithm for minimizing discrete convic functions in fixed dimension

TL;DR

This paper introduces a polynomial-time algorithm for minimizing discrete convic functions in fixed dimensions, significantly improving the efficiency of solving such optimization problems.

Contribution

The paper presents the first polynomial algorithm for minimizing discrete convic functions in fixed dimension, using a bounded number of comparison oracle calls.

Findings

The algorithm uses $2^{O(n^2 ext{log} n)} ext{log} ho$ comparison calls.

It has $2^{O(n^2 ext{log} n)} ext{poly}( ext{log} ho)$ bit complexity.

This resolves the open problem of polynomial-time minimization for discrete convic functions.

Abstract

Recently classes of conic and discrete conic functions were introduced. In this paper we use the term convic instead conic. The class of convic functions properly includes the classes of convex functions, strictly quasiconvex functions and the class of quasiconvex polynomials. On the other hand, the class of convic functions is properly included in the class of quasiconvex functions. The discrete convic function is a discrete analogue of the convic function. Recently the lower bound $3^{n-1}\log (2 \rho-1)$ for the number of calls to the comparison oracle needed to find the minimum of the discrete convic function defined on integer points of some $n$-dimensional ball with radius $\rho$ was obtained. But the problem of the existence of a polynomial (in $\log\rho$ for fixed $n$) algorithm for minimizing such functions has remained open. In this paper, we answer positively the question of the existence of such an algorithm. Namely, we propose an algorithm for minimizing discrete convic functions that uses $2^{O(n^2 \log n)} \log \rho$ calls to the comparison oracle and has $2^{O(n^2 \log n)} \mbox{poly }(\log \rho)$ bit complexity.