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

TL;DR

This paper introduces a fast branch-and-bound algorithm for globally solving the Tikhonov regularized total least squares problem by efficiently estimating bounds, outperforming existing bisection methods in speed and accuracy.

Contribution

The paper proposes a novel branch-and-bound algorithm that guarantees global optimality for the problem, improving upon existing bisection methods with a new underestimation technique.

Findings

The new algorithm achieves a global . -approximate solution efficiently.

It outperforms the improved bisection heuristic in computational speed.

The method has a complexity of O(n^3/) for -approximate solutions.

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 novel 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/\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.