trust {dMod}

Non-Linear Optimization

Description

This function carries out a minimization or maximization of a function using a trust region algorithm. See the references for details.

Usage

trust(
  objfun,
  parinit,
  rinit,
  rmax,
  parscale,
  iterlim = 100,
  fterm = sqrt(.Machine$double.eps),
  mterm = sqrt(.Machine$double.eps),
  minimize = TRUE,
  blather = FALSE,
  parupper = Inf,
  parlower = -Inf,
  printIter = FALSE,
  ...
)

trustL1(
  objfun,
  parinit,
  mu = 0 * parinit,
  one.sided = FALSE,
  lambda = 1,
  rinit,
  rmax,
  parscale,
  iterlim = 100,
  fterm = sqrt(.Machine$double.eps),
  mterm = sqrt(.Machine$double.eps),
  minimize = TRUE,
  blather = FALSE,
  blather2 = FALSE,
  parupper = Inf,
  parlower = -Inf,
  printIter = FALSE,
  ...
)

Arguments

objfun

an R function that computes value, gradient, and Hessian of the function to be minimized or maximized and returns them as a list with components value, gradient, and hessian. Its first argument should be a vector of the length of parinit followed by any other arguments specified by the ... argument.

parinit

starting parameter values for the optimization. Must be feasible (in the domain).

rinit

starting trust region radius. The trust region radius (see details below) is adjusted as the algorithm proceeds. A bad initial value wastes a few steps while the radius is adjusted, but does not keep the algorithm from working properly.

rmax

maximum allowed trust region radius. This may be set very large. If set small, the algorithm traces a steepest descent path (steepest ascent, when minimize = FALSE).

parscale

an estimate of the size of each parameter at the minimum. The algorithm operates as if optimizing function(x, ...) objfun(x / parscale, ...). May be missing in which case no rescaling is done. See also the details section below.

iterlim

a positive integer specifying the maximum number of iterations to be performed before the program is terminated.

fterm

a positive scalar giving the tolerance at which the difference in objective function values in a step is considered close enough to zero to terminate the algorithm.

mterm

a positive scalar giving the tolerance at which the two-term Taylor-series approximation to the difference in objective function values in a step is considered close enough to zero to terminate the algorithm.

minimize

If TRUE minimize. If FALSE maximize.

blather

If TRUE return extra info.

parupper

named numeric vector of upper bounds.

parlower

named numeric vector of lower bounds.

printIter

print iteration information to R console

...

additional argument to objfun

mu

named numeric value. The reference value for L1 penalized parameters.

one.sided

logical. One-sided penalization.

lambda

strength of the L1 penalty

blather2

even more information

Details

See Fletcher (1987, Section 5.1) or Nocedal and Wright (1999, Section 4.2) for detailed expositions.

Value

A list containing the following components:

• value: the value returned by objfun at the final iterate.

• gradient: the gradient returned by objfun at the final iterate.

• hessian: the Hessian returned by objfun at the final iterate.

• argument: the final iterate

• converged: if TRUE the final iterate was deemed optimal by the specified termination criteria.

• iterations: number of trust region subproblems done (including those whose solutions are not accepted).

• argpath: (if blather == TRUE) the sequence of iterates, not including the final iterate.

• argtry: (if blather == TRUE) the sequence of solutions of the trust region subproblem.

• steptype: (if blather == TRUE) the sequence of cases that arise in solutions of the trust region subproblem. "Newton" means the Newton step solves the subproblem (lies within the trust region). Other values mean the subproblem solution is constrained. "easy-easy" means the eigenvectors corresponding to the minimal eigenvalue of the rescaled Hessian are not all orthogonal to the gradient. The other cases are rarely seen. "hard-hard" means the Lagrange multiplier for the trust region constraint is minus the minimal eigenvalue of the rescaled Hessian; "hard-easy" means it isn't.

• accept: (if blather == TRUE) indicates which of the sequence of solutions of the trust region subproblem were accepted as the next iterate. (When not accepted the trust region radius is reduced, and the previous iterate is kept.)

• r: (if blather == TRUE) the sequence of trust region radii.

• rho: (if blather == TRUE) the sequence of ratios of actual over predicted decrease in the objective function in the trust region subproblem, where predicted means the predicted decrease in the two-term Taylor series model used in the subproblem.

• valpath: (if blather == TRUE) the sequence of objective function values at the iterates.

• valtry: (if blather == TRUE) the sequence of objective function values at the solutions of the trust region subproblem.

• preddiff: (if blather == TRUE) the sequence of predicted differences using the two-term Taylor-series model between the function values at the current iterate and at the solution of the trust region subproblem.

• stepnorm: (if blather == TRUE) the sequence of norms of steps, that is distance between current iterate and proposed new iterate found in the trust region subproblem.