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An Algorithm Based on Resolvant Operators for Solving Positively Semidefinite Variational Inequalities

Abstract

A new monotonicity, -monotonicity, is introduced, and the resolvant operator of an -monotone operator is proved to be single-valued and Lipschitz continuous. With the help of the resolvant operator, the positively semidefinite general variational inequality (VI) problem VI is transformed into a fixed point problem of a nonexpansive mapping. And a proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method is given for calculating -solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable.

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Correspondence to Juhe Sun.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Sun, J., Zhang, S. & Zhang, L. An Algorithm Based on Resolvant Operators for Solving Positively Semidefinite Variational Inequalities. Fixed Point Theory Appl 2007, 076040 (2007). https://doi.org/10.1155/2007/76040

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  • DOI: https://doi.org/10.1155/2007/76040

Keywords

  • Variational Inequality
  • Differential Geometry
  • Computational Biology
  • Resolvant Operator