Communications in Mathematical Sciences

Volume 13 (2015)

Number 8

Density matrix minimization with ${\ell}_1$ regularization

Pages: 2097 – 2117



Rongjie Lai (Department of Mathematics, Rensselaer Polytechnic Institute, Troy, New York, U.S.A.)

Jianfeng Lu (Departments of Mathematics, Physics, and Chemistry, Duke University, Durham, North Carolina, U.S.A.)

Stanley Osher (Department of Mathematics and Institute for Pure and Applied Mathematics, University of California at Los Angeles)


We propose a convex variational principle to find sparse representation of low-lying eigenspace of symmetric matrices. In the context of electronic structure calculation, this corresponds to a sparse density matrix minimization algorithm with ${\ell}_1$ regularization. The minimization problem can be efficiently solved by a split Bregman iteration type algorithm. We further prove that from any initial condition, the algorithm converges to a minimizer of the variational principle.


density matrix, ${\ell}_1$ regularization, eigenspace

2010 Mathematics Subject Classification

65F30, 65K10

Published 3 September 2015