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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(ε -2 ) to drive the norm of the gradient below ε. This shows that the upper bound of O(ε -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(ε -3/2 ) evaluations known for cubically regularized Newton’s methods is also shown to be tight.
MSC:
90C30Nonlinear programming
65K05Mathematical programming (numerical methods)
49M37Methods of nonlinear programming type in calculus of variations
49M15Newton-type methods in calculus of variations
49M05Numerical methods in calculus of variations based on necessary conditions
58C15Implicit function theorems and global Newton methods on manifolds
90C60Abstract computational complexity for mathematical programming problems
68Q25Analysis of algorithms and problem complexity