Updating the regularization parameter in the adaptive cubic regularization algorithm

The adaptive cubic regularization method (Cartis et al. in Math. Program. Ser. A 127(2):245–295, 2011; Math. Program. Ser. A. 130(2):295–319, 2011) has been recently proposed for solving unconstrained minimization problems. At each iteration of this method, the objective function is replaced by a cubic approximation which comprises an adaptive regularization parameter whose role is related to the local Lipschitz constant of the objective’s Hessian. We present new updating strategies for this parameter based on interpolation techniques, which improve the overall numerical performance of the algorithm. Numerical experiments on large nonlinear least-squares problems are provided.