numerical_hessians

Hessians are needed and will be approximated by finite differences

Specification

• Alias: None

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Optional		fd_step_size	Step size used when computing gradients and Hessians
Optional (Choose One)	Step Scaling	relative	(Default) Scale step size by the parameter value
		absolute	Do not scale step-size
		bounds	Scale step-size by the domain of the parameter
Optional (Choose One)	Finite Difference Type	forward	(Default) Use forward differences
		central	Use central differences

Description

The numerical_hessians specification means that Hessian information is needed and will be computed with finite differences using either first-order gradient differencing (for the cases of analytic_gradients or for the functions identified by id_analytic_gradients in the case of mixed_gradients) or first- or second-order function value differencing (all other gradient specifications). In the former case, the following expression

$${\mathrm{\nabla }}^{2}f(\mathbf{x}{)}_{i}ong\frac{\mathrm{\nabla }f(\mathbf{x}+h{\mathbf{e}}_i)-\mathrm{\nabla }f(\mathbf{x})}{h}$$

estimates the $i^{th}$ Hessian column, and in the latter case, the following expressions

$${\mathrm{\nabla }}^{2}f(\mathbf{x}{)}_{i,j}ong‘‘‘‘\frac{f(\mathbf{x}+h_i{\mathbf{e}}_i+h_j{\mathbf{e}}_j)-f(\mathbf{x}+h_i{\mathbf{e}}_i)-f(\mathbf{x}-h_j{\mathbf{e}}_j)+f(\mathbf{x})}{h_i{h}_j}$$

and

$${\mathrm{\nabla }}^{2}f(\mathbf{x}{)}_{i,j}ong‘‘‘‘\frac{f(\mathbf{x}+h{\mathbf{e}}_i+h{\mathbf{e}}_j)-f(\mathbf{x}+h{\mathbf{e}}_i-h{\mathbf{e}}_j)-f(\mathbf{x}-h{\mathbf{e}}_i+h{\mathbf{e}}_j)+f(\mathbf{x}-h{\mathbf{e}}_i-h{\mathbf{e}}_j)}{4h^2}$$

provide first- and second-order estimates of the $i{j}^{th}$ Hessian term. Prior to Dakota 5.0, Dakota always used second-order estimates. In Dakota 5.0 and newer, the default is to use first-order estimates (which honor bounds on the variables and require only about a quarter as many function evaluations as do the second-order estimates), but specifying central after numerical_hessians causes Dakota to use the old second-order estimates, which do not honor bounds. In optimization algorithms that use Hessians, there is little reason to use second-order differences in computing Hessian approximations.