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 0 -norm involved. In this paper, we resolv...

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Bibliographic Details
Published in:Mathematical programming Vol. 195; no. 1-2; pp. 903 - 928
Main Authors: Zhao, Chen, Xiu, Naihua, Qi, Houduo, Luo, Ziyan
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2022
Springer
Springer Nature B.V
Subjects:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Euler-Lagrange equation
Full Length Paper
Image processing
Iterative methods
Lagrangian function
Machine learning
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Nonlinear equations
Nonlinear programming
Numerical Analysis
Signal processing
Theoretical
The Newton method
90C30
90C46
Locally quadratic convergence
Sparse nonlinear programming
49M15
Lagrangian equation
Application
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary: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 0 -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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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-021-01719-x