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 ǫ-approximation solution in a computational effort of at most O(n 3 / √ ǫ) under the same assumption as in the bisection method. The numerical results demonstrate that our new global
We study the n-dimensional problem of finding the smallest ball enclosing the intersection of p given balls, the so-called Chebyshev center problem (CC B ). It is a minimax optimization problem and the inner maximization is a uniform quadratic optimization problem (UQ). When p ≤ n, (UQ) is known to enjoy a strong duality and consequently (CC B ) is solved via a standard convex quadratic programming (SQP). In this paper, we first prove that (CC B ) is NP-hard and the special case when n = 2 is strongly polynomially solvable. With the help of a newly introduced linear programming relaxation (LP), the (SQP) relaxation is reobtained more directly and the first approximation bound for the solution obtained by (SQP) is established for the hard case p > n. Finally, also based on (LP), we show that (CC B ) is polynomially solvable when either n or p − n(> 0) is fixed.
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