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A fast algorithm for globally solving Tikhonov regularized total least squares problem

  • Yong Xia
  • Longfei Wang
  • Meijia Yang
Article
  • 30 Downloads

Abstract

The total least squares problem with the general Tikhonov regularization can be reformulated as a one-dimensional parametric minimization problem (PM), where each parameterized function evaluation corresponds to solving an n-dimensional trust region subproblem. Under a mild assumption, the parametric function is differentiable and then an efficient bisection method has been proposed for solving (PM) in literature. In the first part of this paper, we show that the bisection algorithm can be greatly improved by reducing the initially estimated interval covering the optimal parameter. It is observed that the bisection method cannot guarantee to find the globally optimal solution since the nonconvex (PM) could have a local non-global minimizer. The main contribution of this paper is to propose an efficient branch-and-bound algorithm for globally solving (PM), based on a new underestimation of the parametric function over any given interval using only the information of the parametric function evaluations at the two endpoints. We can show that the new algorithm (BTD Algorithm) returns a global \(\epsilon \)-approximation solution in a computational effort of at most \(O(n^3/\sqrt{\epsilon })\) under the same assumption as in the bisection method. The numerical results demonstrate that our new global optimization algorithm performs even much faster than the improved version of the bisection heuristic algorithm.

Keywords

Total least squares Tikhonov regularization Trust region subproblem Fractional program Lower bound Branch and bound 

Mathematics Subject Classification

65F20 90C26 90C32 90C20 

Notes

Acknowledgements

The authors are grateful to the two anonymous referees for their valuable comments and suggestions.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.State Key Laboratory of Software Development Environment, LMIB of the Ministry of Education, School of Mathematics and System SciencesBeihang UniversityBeijingPeople’s Republic of China

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