Finding the Global Optimum of a Class of Quartic Minimization Problem

TL;DR

This paper analyzes a specific nonconvex quartic minimization problem with a spherical constraint, demonstrating that certain algorithms can reliably find the global minimum, with applications to Bose-Einstein condensates.

Contribution

The paper characterizes the problem as a nonlinear eigenvalue problem with a unique nonnegative eigenvector that is the global minimizer, and proves global convergence of an inexact ADMM method.

Findings

Inexact ADMM converges globally to the optimal solution.

The problem's structure ensures the nonnegative eigenvector is the global minimizer.

Numerical experiments validate the theoretical results.

Abstract

We consider a special nonconvex quartic minimization problem over a single spherical constraint, which includes the discretized energy functional minimization problem of non-rotating Bose-Einstein condensates (BECs) as one of the important applications. Such a problem is studied by exploiting its characterization as a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv), which admits a unique nonnegative eigenvector, and this eigenvector is exactly the global minimizer to the quartic minimization. With these properties, any algorithm converging to the nonnegative stationary point of this optimization problem finds its global minimum, such as the regularized Newton (RN) method. In particular, we obtain the global convergence to global optimum of the inexact alternating direction method of multipliers (ADMM) for this problem. Numerical experiments for applications in non-rotating BEC validate our theories.