Smooth and Semismooth Newton Methods for Constrained Approximation and Estimation |
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| Authors: | Hongxia Yin Chen Ling Liqun Qi |
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| Institution: | 1. Department of Mathematics and Statistics , Minnesota State University , Mankato , Minnesota , USA;2. CAS Research Center of Fictional Economy and Data Science , Beijing , P. R. China hongxia.yin@mnsu.edu;4. School of Science , Hangzhou Dianzi University , Hangzhou , P. R. China;5. Department of Applied Mathematics , The Hong Kong Polytechnic University , Hung Hom , Kowloon , Hong Kong |
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| 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. |
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| Keywords: | Convergence Globalized Newton method Hölder continuity L 2 approximation Positive polynomial interpolation Probability density estimation Semismoothness |
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