newton: Generalized newton optimizer.

In TMB: Template Model Builder: A General Random Effect Tool Inspired by 'ADMB'

Generalized newton optimizer.

Description

Generalized newton optimizer used for the inner optimization problem.

Usage

newton(
  par,
  fn,
  gr,
  he,
  trace = 1,
  maxit = 100,
  tol = 1e-08,
  alpha = 1,
  smartsearch = TRUE,
  mgcmax = 1e+60,
  super = TRUE,
  silent = TRUE,
  ustep = 1,
  power = 0.5,
  u0 = 1e-04,
  grad.tol = tol,
  step.tol = tol,
  tol10 = 0.001,
  env = environment(),
  ...
)

Arguments

par	Initial parameter.
fn	Objective function.
gr	Gradient function.
he	Sparse hessian function.
trace	Print tracing information?
maxit	Maximum number of iterations.
tol	Convergence tolerance.
alpha	Newton stepsize in the fixed stepsize case.
smartsearch	Turn on adaptive stepsize algorithm for non-convex problems?
mgcmax	Refuse to optimize if the maximum gradient component is too steep.
super	Supernodal Cholesky?
silent	Be silent?
ustep	Adaptive stepsize initial guess between 0 and 1.
power	Parameter controlling adaptive stepsize.
u0	Parameter controlling adaptive stepsize.
grad.tol	Gradient convergence tolerance.
step.tol	Stepsize convergence tolerance.
tol10	Try to exit if last 10 iterations not improved more than this.
env	Environment for cached Cholesky factor.
...	Currently unused.

Details

If smartsearch=FALSE this function performs an ordinary newton optimization on the function fn using an exact sparse hessian function. A fixed stepsize may be controlled by alpha so that the iterations are given by:

u_{n+1} = u_n - \alpha f''(u_n)^{-1}f'(u_n)

If smartsearch=TRUE the hessian is allowed to become negative definite preventing ordinary newton iterations. In this situation the newton iterations are performed on a modified objective function defined by adding a quadratic penalty around the expansion point u_0:

f_{t}(u) = f(u) + \frac{t}{2} \|u-u_0\|^2

This function's hessian ( f''(u)+t I ) is positive definite for t sufficiently large. The value t is updated at every iteration: If the hessian is positive definite t is decreased, otherwise increased. Detailed control of the update process can be obtained with the arguments ustep, power and u0.

Value

List with solution similar to optim output.

See Also

newtonOption

TMB documentation built on Sept. 11, 2024, 7:06 p.m.