The paper is concerned with a general cubic regularization framework for unconstrained optimization which has roots in earlier algorithms. It contains a thorough introduction to relevant contributions and presents an appropriate list of references on this subject. The authors consider the convergence properties. The framework allows for the approximate solution of the key step calculation. Preliminary numerical experiments with small-scale problems are reported. For Part II see [Math. Program. 130, No. 2 (A), 295‒319 (2011; Zbl 1229.90193)].
Reviewer: Francisco Guerra Vazquez (Puebla)