In each iteration , the (dual) quasiNewton, hybrid quasiNewton, conjugate gradient, and NewtonRaphson minimization techniques use iterative linesearch algorithms that try to optimize a linear, quadratic, or cubic approximation of along a feasible descent search direction

by computing an approximately optimal scalar .
Therefore, a linesearch algorithm is an iterative process that optimizes a nonlinear function of one parameter () within each iteration of the optimization technique, which itself tries to optimize a linear or quadratic approximation of the nonlinear objective function of parameters . Since the outside iteration process is based only on the approximation of the objective function, the inside iteration of the linesearch algorithm does not have to be perfect. Usually, the choice of significantly reduces (in a minimization) the objective function. Criteria often used for termination of linesearch algorithms are the Goldstein conditions (refer to Fletcher (1987)).
Various linesearch algorithms can be selected using the LINESEARCH= option. The linesearch method LINESEARCH=2 seems to be superior when function evaluation consumes significantly less computation time than gradient evaluation. Therefore, LINESEARCH=2 is the default value for NewtonRaphson, (dual) quasiNewton, and conjugate gradient optimizations.
A special default linesearch algorithm for TECH=HYQUAN is useful only for least squares problems and cannot be chosen by the LINESEARCH= option. This method uses three columns of the Jacobian matrix, which for large can require more memory than using the algorithms designated by LINESEARCH=1 through LINESEARCH=8.
The linesearch methods LINESEARCH=2 and LINESEARCH=3 can be modified to exact line search by using the LSPRECISION= option (specifying the parameter in Fletcher (1987)). The linesearch methods LINESEARCH=1, LINESEARCH=2, and LINESEARCH=3 satisfy the lefthandside and righthandside Goldstein conditions (refer to Fletcher (1987)). When derivatives are available, the linesearch methods LINESEARCH=6, LINESEARCH=7, and LINESEARCH=8 try to satisfy the righthandside Goldstein condition; if derivatives are not available, these linesearch algorithms use only function calls.