Line-Search Methods

The NLP Procedure

Line-Search Methods

In each iteration k, the (dual) quasi-Newton, hybrid 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 s^(k)

$x^{(k+1)} = x^{(k)} + \alpha^{(k)} s^{(k)} , \alpha^{(k)} \gt 0$

by computing an approximately optimal scalar $\alpha^{(k)}$ .

Therefore, a line-search algorithm is an iterative process that optimizes a nonlinear function $f=f(\alpha)$ of one parameter ( $\alpha$ ) within each iteration k of the optimization technique, which itself tries to optimize a linear or quadratic approximation of the nonlinear objective function f=f(x) of n parameters x. 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, the choice of $\alpha$ significantly reduces (in a minimization) the objective function. Criteria often used for termination of line-search algorithms are the Goldstein conditions (refer to Fletcher 1987).

Various line-search algorithms can be selected by using the LIS= option. The line-search method LIS=2 seems to be superior when function evaluation consumes significantly less computation time than gradient evaluation. Therefore, LIS=2 is the default value for Newton-Raphson, (dual) quasi-Newton, and conjugate gradient optimizations.

A special default line-search algorithm for TECH= HYQUAN is useful only for least-squares problems and cannot be chosen by the LIS= option. This method uses three columns of the m ×n Jacobian matrix, which can for large m require more memory than using the algorithms designated by LIS=1 through LIS=8.

The line-search methods LIS=2 and LIS=3 can be modified to exact line-search by using the LSPRECISION= option (specifying the $\sigma$ parameter in Fletcher, 1987). The line-search methods LIS=1, LIS=2, and LIS=3 satisfy the left-hand side and right-hand side Goldstein conditions (refer to Fletcher 1987). When derivatives are available, the line-search methods LIS=6, LIS=7,and LIS=8 try to satisfy the right-hand side Goldstein condition; if derivatives are not available, these line-search algorithms use only function calls.

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