We show that the steepest-descent and Newton's methods for unconstrained nonconvex optimization under standard assumptions may both require a number of iterations and function evaluations arbitrarily close to the steepest-descent's global worst-case complexity bound. This shows that the latter upper bound is essentially tight for steepest descent and that Newton's method may be as slow as the steepest-descent method in the worst case. Then the cubic regularization of Newton's method (Griewank (1981), Nesterov & Polyak (2006)) is considered and extended to large-scale problems, while preserving the same order of its improved worst-case complexity (by comparison to that of steepest-descent); this improved worst-case bound is also shown to be essentially tight. We further show that the cubic regularization approach is, in fact, optimal from a worst-case complexity point of view amongst a class of second-order methods. The worst-case problem-evaluation complexity of constrained optimization will also be discussed, time permitting. This is joint work with Nick Gould (Rutherford Appleton Laboratory, UK) and Philippe Toint (Namur University, Belgium).
Group for Research in Decision Analysis