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.
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The authors are grateful to the two anonymous referees for their valuable comments and suggestions.
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This research was supported by NSFC under Grants 11571029, 11471325 and 11771056, and by fundamental research funds for the Central Universities under Grant YWF-18-BJ-Y-16.
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Xia, Y., Wang, L. & Yang, M. A fast algorithm for globally solving Tikhonov regularized total least squares problem. J Glob Optim 73, 311–330 (2019). https://doi.org/10.1007/s10898-018-0719-x
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DOI: https://doi.org/10.1007/s10898-018-0719-x
Keywords
- Total least squares
- Tikhonov regularization
- Trust region subproblem
- Fractional program
- Lower bound
- Branch and bound