Minimizing Convex Functions with Rational Minimizers

Abstract

Given a separation oracle SO for a convex function f that has an integral minimizer inside a box with radius R, we show how to e ciently nd a minimizer of f using at most O(n(n+log(R))) calls to SO. When the set of minimizers of f has integral extreme points, our algorithm outputs an integral minimizer of f . This improves upon the previously best oracle complexity of O(n 2 (n + log(R))) obtained by an elegant application of simultaneous diophantine approximation due to Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. We conjecture that our oracle complexity is tight up to constant factors.Our result immediately implies a strongly polynomial algorithm for the Submodular Function Minimization problem that makes at most O(n 3 ) calls to an evaluation oracle. This improves upon the previously best O(n 3 log 2 (n)) oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, Végh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity O(n 3 log(n)) given in the former work, answering two open problems posted therein.Our result is achieved by an application of the LLL algorithm [Lenstra, Lenstra and Lovász, Math. Ann. 1982] for the shortest lattice vector problem. We show how an approximately shortest vector of certain lattice can be used to reduce the dimension of the problem, and how the oracle complexity of such a procedure is advantageous compared with the Grötschel-Lovász-Schrijver approach that uses simultaneous diophantine approximation. Our analysis of the oracle complexity is based on a potential function that captures simultaneously the size of the search set and the density of the lattice. To achieve the O(n 2 ) term in the oracle complexity, technical ingredients from convex geometry are applied.

Minimizing Convex Functions with Integral Minimizers

“…The following Theorem 1.2 is obtained by directly applying Theorem 1.1 to the Lovász extension f of the function f , together with the well-known fact that a separation oracle for f can be implemented using n calls to the evaluation oracle ([31, Theorem 61]). We provide details on these definitions and the proof of Theorem 1.2 in the full version [25].…”

“…Approach: Previous O(n 3 ) Oracle Complexity For the moment, let's take K to be an ellipsoid. Such an ellipsoid can be obtained by Vaidya's volumetric center cutting plane method 5 [47] (see the full version [25] for a statement of the theorem). One natural idea in finding the hyperplane comes from the following geometric intuition: when the ellipsoid K is "flat" enough in one direction, then all its integral points lie on a hyperplane P .…”

“…We believe this result, while having an extraneous factor of log(n), is elegant and interesting in its own right. The details for this algorithm and its analysis are given in the full version [25].…”

“…Prior to our work, an elegant application of simultaneous diophantine approximation due to Grötschel, Lovász and Schrijver [19,20] gives 2 a strongly polynomial algorithm that finds the minimizer of f using O(n 2 (n + log(R))) calls 3 to the separation oracle. We refer interested readers to [20,Chapter 6] for a detailed presentation of their approach (also see the full version [25]). The purpose of the present paper is to design a strongly polynomial algorithm with an improved number of separation oracle calls.…”

Stochastic Localization Methods for Convex Discrete Optimization via Simulation

“…On the other hand, the number of arithmetic operations in each iteration is mainly determined by the answer to the second problem, since other parts of the algorithm requires polynomially many arithmetic operations. Recently, the two problems are solved in Jiang (2020). For the first problem, the author has designed an alternative potential function that includes both the volume of search polytope and the density of integral points in the current subspace.…”