On large-scale unconstrained optimization problems and higher order methods

Abstract

Third-order methods will, in most cases, use fewer iterations than a second-order method to reach the same accuracy. However, the number of arithmetic operations per iteration is higher for third-order methods than for a second-order method. Newton's method is the most commonly used second order method and Halley's method is the most well-known third-order method. Newton's method is more used in practical applications than any third-order method. We will show that for a large class of problems, the ratio of the number of arithmetic operations of Halley's method and Newton's method is constant per iteration. It is shown that

We show that the zero elements in the Hessian matrix induce zero elements in the tensor (third derivative). The sparsity structure in the Hessian matrix we consider is the skyline or envelope structure. This is a very convenient structure for solving linear systems of equations with a direct method. The class of matrices that have a skyline structure includes banded and dense matrices. Numerical testing confirms that the ratio of the number of arithmetic operations of a third-order method and Newton's method is constant per iteration, and is independent of the number of unknowns.

Keywords:

• unconstrained optimization problems
• induced sparsity
• higher order methods
• tensor computations

AMS Subject Classification Codes :

• 65K05
• 65F50
• 90C30
• 90C06

Acknowledgements

We would like to thank the anonymous referee for careful evaluation of this paper and Hubert Schwetlick for pointing out the early history of the Halley class.