In each iteration k, the (dual) quasi-Newton, conjugate gradient, and Newton-Raphson minimization techniques use iterative line-search algorithms
that try to optimize a linear, quadratic, or cubic approximation of f along a feasible descent search direction ,
by computing an approximately optimal scalar .
Therefore, a line-search algorithm is an iterative process that optimizes a nonlinear function of one parameter (
) within each iteration k of the optimization technique. Since the outside iteration process is based only on the approximation of the objective function,
the inside iteration of the line-search algorithm does not have to be perfect. Usually, it is satisfactory that the choice
of
significantly reduces (in a minimization) the objective function. Criteria often used for termination of line-search algorithms
are the Goldstein conditions (see Fletcher 1987).
You can select various line-search algorithms by specifying the LINESEARCH= option. The line-search method LINESEARCH= 2 seems to be superior when function evaluation consumes significantly less computation time than gradient evaluation. Therefore, LINESEARCH= 2 is the default method for Newton-Raphson, (dual) quasi-Newton, and conjugate gradient optimizations.
You can modify the line-search methods LINESEARCH=
2 and LINESEARCH=
3 to be exact line searches by using the LSPRECISION=
option and specifying the parameter described in Fletcher (1987). The line-search methods LINESEARCH=
1, LINESEARCH=
2, and LINESEARCH=
3 satisfy the left-side and right-side Goldstein conditions (see Fletcher 1987). When derivatives are available, the line-search methods LINESEARCH=
6, LINESEARCH=
7, and LINESEARCH=
8 try to satisfy the right-side Goldstein condition; if derivatives are not available, these line-search algorithms use only
function calls.