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newton

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 - α 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) + 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.