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A Lagrange-Newton algorithm for sparse nonlinear programming

A Lagrange-Newton algorithm for sparse nonlinear programming
A Lagrange-Newton algorithm for sparse nonlinear programming

The sparse nonlinear programming (SNP) problem has wide applications in signal and image processing, machine learning and finance, etc. However, the computational challenge posed by SNP has not yet been well resolved due to the nonconvex and discontinuous ℓ-norm involved. In this paper, we resolve this numerical challenge by developing a fast Newton-type algorithm. As a theoretical cornerstone, we establish a first-order optimality condition for SNP based on the concept of strong β-Lagrangian stationarity via the Lagrangian function, and reformulate it as a system of nonlinear equations called the Lagrangian equations. The nonsingularity of the corresponding Jacobian is discussed, based on which the Lagrange–Newton algorithm (LNA) is then proposed. Under mild conditions, we establish the locally quadratic convergence and its iterative complexity estimation. To further demonstrate the efficiency and superiority of our proposed algorithm, we apply LNA to two specific problems arising from compressed sensing and sparse high-order portfolio selection, in which significant benefits accrue from the restricted Newton step.

Application, Lagrangian equation, Locally quadratic convergence, Sparse nonlinear programming, The Newton method
0025-5610
Chen, Zhao
a8bcc4da-4b5c-4966-b699-34d04cb4309a
Xiu, Naihua
8b5770f7-ae35-4dbe-884a-02fb4ea27bee
Qi, Hou-Duo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Lu, Ziyan
b4705c79-7c03-4584-b057-7b492ec00b1c
Chen, Zhao
a8bcc4da-4b5c-4966-b699-34d04cb4309a
Xiu, Naihua
8b5770f7-ae35-4dbe-884a-02fb4ea27bee
Qi, Hou-Duo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Lu, Ziyan
b4705c79-7c03-4584-b057-7b492ec00b1c

Chen, Zhao, Xiu, Naihua, Qi, Hou-Duo and Lu, Ziyan (2021) A Lagrange-Newton algorithm for sparse nonlinear programming. Mathematical Programming. (doi:10.1007/s10107-021-01719-x).

Record type: Article

Abstract

The sparse nonlinear programming (SNP) problem has wide applications in signal and image processing, machine learning and finance, etc. However, the computational challenge posed by SNP has not yet been well resolved due to the nonconvex and discontinuous ℓ-norm involved. In this paper, we resolve this numerical challenge by developing a fast Newton-type algorithm. As a theoretical cornerstone, we establish a first-order optimality condition for SNP based on the concept of strong β-Lagrangian stationarity via the Lagrangian function, and reformulate it as a system of nonlinear equations called the Lagrangian equations. The nonsingularity of the corresponding Jacobian is discussed, based on which the Lagrange–Newton algorithm (LNA) is then proposed. Under mild conditions, we establish the locally quadratic convergence and its iterative complexity estimation. To further demonstrate the efficiency and superiority of our proposed algorithm, we apply LNA to two specific problems arising from compressed sensing and sparse high-order portfolio selection, in which significant benefits accrue from the restricted Newton step.

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More information

Accepted/In Press date: 5 October 2021
e-pub ahead of print date: 21 October 2021
Published date: 21 October 2021
Additional Information: Funding Information: This research was partially supported by the National Natural Science Foundation of China (11771038, 11971052, 12011530155) and Beijing Natural Science Foundation (Z190002) Publisher Copyright: © 2021, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.
Keywords: Application, Lagrangian equation, Locally quadratic convergence, Sparse nonlinear programming, The Newton method

Identifiers

Local EPrints ID: 452474
URI: http://eprints.soton.ac.uk/id/eprint/452474
ISSN: 0025-5610
PURE UUID: 971f7f01-16c6-4aec-b017-a43e49224e22
ORCID for Hou-Duo Qi: ORCID iD orcid.org/0000-0003-3481-4814

Catalogue record

Date deposited: 11 Dec 2021 11:07
Last modified: 06 Feb 2023 19:27

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Contributors

Author: Zhao Chen
Author: Naihua Xiu
Author: Hou-Duo Qi ORCID iD
Author: Ziyan Lu

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