A Fast and Convergent Algorithm for Unassigned Distance Geometry Problems

TL;DR

This paper introduces a fast, convergent algorithm for unassigned distance geometry problems using a novel quadratic measurement model with $\u2113_0$-norm, and demonstrates its effectiveness through theoretical convergence proofs and numerical validation.

Contribution

It presents a new quadratic measurement model with $\u2113_0$-norm and a fast iterative hard thresholding algorithm with proven convergence for uDGP.

Findings

Algorithm converges to an L-stationary point.

Outperforms existing $\u2113_1$-norm-based methods.

Validated on turnpike and beltway problems.

Abstract

In this paper, we propose a fast and convergent algorithm to solve unassigned distance geometry problems (uDGP). Technically, we construct a novel quadratic measurement model by leveraging $\ell_0$-norm instead of $\ell_1$-norm in the literature. To solve the nonconvex model, we establish its optimality conditions and develop a fast iterative hard thresholding (IHT) algorithm. Theoretically, we rigorously prove that the whole generated sequence converges to the L-stationary point with the help of the Kurdyka-Lojasiewicz (KL) property. Numerical studies on the turnpike and beltway problems validate its superiority over existing $\ell_1$-norm-based method.