On computing with some convex relaxations for the maximum-entropy sampling problem

TL;DR

This paper introduces a new convex relaxation bound for the NP-hard maximum-entropy sampling problem, demonstrating its invariance, and proposes a variable-fixing method for exact solutions, with successful experimental results.

Contribution

It presents a generalized factorization-based bound for CMESP, invariant under scaling, and develops a variable-fixing approach for branch-and-bound algorithms.

Findings

The factorization bound is invariant under scaling.

The variable-fixing method can be integrated into branch-and-bound schemes.

Combining bounds with the 'mixing' technique improves solution quality.

Abstract

Based on a factorization of an input covariance matrix, we define a mild generalization of an upper bound of Nikolov (2015) and Li and Xie (2020) for the NP-Hard constrained maximum-entropy sampling problem (CMESP). We demonstrate that this factorization bound is invariant under scaling and also independent of the particular factorization chosen. We give a variable-fixing methodology that could be used in a branch-and-bound scheme based on the factorization bound for exact solution of CMESP. We report on successful experiments with a commercial nonlinear-programming solver. We further demonstrate that the known "mixing" technique can be successfully used to combine the factorization bound with the factorization bound of the complementary CMESP, and also with the "linx bound" of Anstreicher (2020).