Minimizing a Low-Dimensional Convex Function Over a High-Dimensional Cube

Abstract

For a matrix , , and a convex function , we are interested in minimizing over the set . We will study separable convex functions and sharp convex functions . Moreover, the matrix is unknown to us. Only the number of rows and are revealed. The composite function is presented by a zeroth and first order oracle only. Our main result is a proximity theorem that ensures that an integral minimum and a continuous minimum for separable convex and sharp convex functions are always “close” by. This will be a key ingredient in developing an algorithm for detecting an integer minimum that achieves a running time of roughly . In the special case when is given explicitly and is separable convex one can also adapt an algorithm of Hochbaum and Shanthikumar [J. ACM, 37 (1990), pp. 843–862]. The running time of this adapted algorithm matches the running time of our general algorithm.