Minimizing Convex Functions with Rational Minimizers

Given a separation oracle SO for convex function f defined on ℝ n that has an integral minimizer inside box with radius R , we show how to find exact of using at most O(n (n log (n)/ (n) + ( ))) calls and poly )) arithmetic operations, or (nR) exp O(n) ) ⋅ (log (R) operations. When the set minimizers extreme points, our algorithm outputs . This improves upon previously best complexity 2 polynomial time algorithms nR exponential obtained by [Grötschel, Lovász Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. Our improvement Grötschel, Schrijver’s result generalizes setting where is rational polyhedron bounded vertex complexity. For Submodular Function Minimization problem, immediately implies strongly makes 3 )/log evaluation oracle, oracle. These improve given in [Lee, Sidford Wong, FOCS 2015] [Dadush, Végh Zambelli, SODA 2018], former work. achieved via reduction Shortest Vector Problem lattices. We approximately shortest vector auxiliary lattice can be used effectively reduce dimension problem. analysis based potential simultaneously captures size search density lattice, which analyze tools from geometry theory.