Finding a Maximal Determinant Principal Submatrix via Hadamard's Inequality and Conic Relaxations

TL;DR

This paper explores exact and approximate methods for finding the principal submatrix with the maximum determinant, introducing bounds and relaxations that improve solution quality and computational efficiency.

Contribution

It proposes a new upper bound based on Hadamard's inequality, a projection scheme for small instances, and semidefinite programming relaxations, advancing the understanding of determinant maximization.

Findings

The projection scheme effectively solves small- to medium-scale problems.

The LP relaxation provides reliable performance evaluation.

Semidefinite programming yields stronger upper bounds.

Abstract

An important yet challenging problem in numerical linear algebra is finding a principal submatrix with the maximum determinant. In this paper, we examine several exact and approximate approaches to this problem. We first propose an upper bound based on Hadamard's inequality, along with a projection scheme based on the Gram-Schmidt process without normalization. This scheme yields a highly effective exact algorithm for solving small- to medium-scale instances. We then study a linear programming (LP) relaxation that facilitates reliable performance evaluation when the exact method returns only near-optimal solutions, and prove that our projection scheme also strengthens the upper bound obtained from the LP relaxation. Finally, we present stronger upper bounds via semidefinite programming, further illustrating the intrinsic difficulty of determinant maximization.