Smooth and semismooth newton methods for constrained approximation and estimation

Hongxia Yin, Chen Ling, Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

1 Citation (Scopus)


In the article, we show that the constrained L2approximation 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.
Original languageEnglish
Pages (from-to)558-589
Number of pages32
JournalNumerical Functional Analysis and Optimization
Issue number5
Publication statusPublished - 1 May 2012


  • Convergence
  • Globalized Newton method
  • Hölder continuity
  • L approximation 2
  • Positive polynomial interpolation
  • Probability density estimation
  • Semismoothness

ASJC Scopus subject areas

  • Analysis
  • Signal Processing
  • Computer Science Applications
  • Control and Optimization


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