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Title: Smooth and semismooth newton methods for constrained approximation and estimation
Authors: Yin, H
Ling, C
Qi, L 
Keywords: Convergence
Globalized Newton method
Hölder continuity
L 2 approximation
Positive polynomial interpolation
Probability density estimation
Issue Date: 2012
Publisher: Taylor & Francis
Source: Numerical functional analysis and optimization, 2012, v. 33, no. 5, p. 558-589 How to cite?
Journal: Numerical functional analysis and optimization 
Abstract: In the article, we show that the constrained L 2 approximation problem, the positive polynomial interpolation, and the density estimation problems can all be reformulated as a system of smooth or semismooth equations by using Lagrange duality theory. The obtained equations contain integral functions of the same form. The differentiability or (strong) semismoothness of the integral functions and the Hölder continuity of the Jacobian of the integral function were investigated. Then a globalized Newton-type method for solving these problems was introduced. Global convergence and numerical tests for estimating probability density functions with wavelet basis were also given. The research in this article not only strengthened the theoretical results in literatures but also provided a possibility for solving the probability density function estimation problem by Newton-type method.
ISSN: 0163-0563
DOI: 10.1080/01630563.2011.653071
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