A Fast and Convergent Algorithm for Unassigned Distance Geometry 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.