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On the complexity of steepest descent, Newton’s and regularized Newton’s methods for nonconvex unconstrained optimization problems. (English) Zbl 1211.90225

Summary: It is shown 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 \(O(\varepsilon^{-2})\) to drive the norm of the gradient below \(\varepsilon\). This shows that the upper bound of \(O(\varepsilon^{-2})\) evaluations known for the steepest descent is tight and that Newton’s method may be as slow as the steepest-descent method in the worst case. The improved evaluation complexity bound of \(O(\varepsilon^{-3/2})\) evaluations known for cubically regularized Newton’s methods is also shown to be tight.

MSC:

90C30 Nonlinear programming
65K05 Numerical mathematical programming methods
49M37 Numerical methods based on nonlinear programming
49M15 Newton-type methods
49M05 Numerical methods based on necessary conditions
58C15 Implicit function theorems; global Newton methods on manifolds
90C60 Abstract computational complexity for mathematical programming problems
68Q25 Analysis of algorithms and problem complexity
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