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Levenberg-Marquardt method for the eigenvalue complementarity problem.

Yuan-yuan Chen1, Yan Gao2

  • 1School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China ; College of Mathematics, Qingdao University, Qingdao 266071, China.

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|December 23, 2014
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Summary
This summary is machine-generated.

This study presents a new method for solving the eigenvalue complementarity problem (EiCP) by reformulating it as differentiable equations. The Levenberg-Marquardt method offers a globally convergent and efficient solution for EiCP.

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Area of Science:

  • Numerical Analysis
  • Mathematical Optimization
  • Applied Mathematics

Background:

  • The eigenvalue complementarity problem (EiCP) is a significant model in mechanics, engineering, and economics.
  • Existing methods for solving EiCP rely on nonsmooth techniques, such as nonsmooth or semismooth Newton methods.

Purpose of the Study:

  • To reformulate the EiCP as a system of continuously differentiable equations.
  • To introduce and analyze the Levenberg-Marquardt method for solving the reformulated EiCP.
  • To demonstrate the global convergence and efficiency of the proposed method.

Main Methods:

  • Reformulation of the EiCP into a system of continuously differentiable equations.
  • Application of the Levenberg-Marquardt algorithm to the differentiable system.
  • Theoretical analysis to prove global convergence under mild assumptions.

Main Results:

  • The proposed Levenberg-Marquardt method is proven to be globally convergent for solving the reformulated EiCP.
  • Numerical experiments confirm the efficiency of the developed method.
  • Extensions of the method are explored.

Conclusions:

  • The Levenberg-Marquardt method provides an effective and globally convergent approach for solving the eigenvalue complementarity problem.
  • The reformulation into differentiable equations simplifies the solution process compared to existing nonsmooth methods.