A unified efficient implementation of trust-region type algorithms for unconstrained optimization

Adaptive cubic regularization (ARC) and trust-region (TR) methods use modified linear systems to compute their steps. The modified systems consist in adding some multiple of the identity matrix (or a well-chosen positive definite matrix) to the Hessian to obtain a sufficiently positive definite linear system, the so called shifted system. This type of system was first proposed by Levenberg and Marquardt. Some trial and error is often involved to obtain a specified value for this shift parameter. We provide an efficient unified implementation to track the shift parameter; our implementation encompasses many ARC and TR variants.