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A Semismooth Newton-based Augmented Lagrangian Algorithm for Density Matrix Least Squares Problems

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Abstract

The density matrix least squares problem arises from the quantum state tomography problem in experimental physics and has many applications in signal processing and machine learning, mainly including the phase recovery problem and the matrix completion problem. In this paper, we first reformulate the density matrix least squares problem as an equivalent convex optimization problem and then design an efficient semismooth Newton-based augmented Lagrangian (Ssnal) algorithm to solve the dual of its equivalent form, in which an inexact semismooth Newton (Ssn) algorithm with superlinear or even quadratic convergence is applied to solve the inner subproblems. Theoretically, the global convergence and locally asymptotically superlinear convergence of the Ssnal algorithm are established under very mild conditions. Computationally, the costs of the Ssn algorithm for solving the subproblem are significantly reduced by making full use of low-rank or high-rank property of optimal solutions of the density matrix least squares problem. In order to verify the performance of our algorithm, numerical experiments conducted on randomly generated quantum state tomography problems and density matrix least squares problems with real data demonstrate that the Ssnal algorithm is more effective and robust than the Qsdpnal solver and several state-of-the-art first-order algorithms.

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Notes

  1. http://euler.nmt.edu/~brian/sdplib/sdplib

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Acknowledgements

The work of Yong-Jin Liu was in part supported by the National Natural Science Foundation of China (Grants No. 11871153 and 12271097) and the Natural Science Foundation of Fujian Province of China (Grant No. 2019J01644).

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Authors and Affiliations

  1. Center for Applied Mathematics of Fujian Province, School of Mathematics and Statistics, Fuzhou University, Fuzhou, 350108, Fujian, People’s Republic of China

    Yong-Jin Liu

  2. School of Mathematics and Statistics, Fuzhou University, Fuzhou, 350108, Fujian, People’s Republic of China

    Jing Yu

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Liu, YJ., Yu, J. A Semismooth Newton-based Augmented Lagrangian Algorithm for Density Matrix Least Squares Problems. J Optim Theory Appl 195, 749–779 (2022). https://doi.org/10.1007/s10957-022-02120-0

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