The NLPC Nonlinear Optimization Solver: Line-Search Method :: SAS/OR(R) 9.2 User's Guide: Mathematical Programming

The NLPC Nonlinear Optimization Solver

Line-Search Method

At each iteration , the conjugate gradient (CONGRA), Newton-type (NEWTYP) and quasi-Newton (QUANEW) optimization techniques use iterative line-search algorithms. These algorithms try to optimize a quadratic or cubic approximation of some merit function along the search direction by computing an approximately optimal step length $\alpha^k$ that is used as follows:

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

A line-search algorithm is an iterative process that optimizes a nonlinear function of one variable $\alpha$ within each iteration of the main optimization algorithm, which itself tries to optimize a quadratic approximation of the nonlinear objective function . 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 appropriate choice of $\alpha$ is one that significantly reduces (in the case of minimization) the objective function value. Criteria often used for termination of line-search algorithms are the Goldstein conditions; see Fletcher (1987).

The line-search method in the NLPC solver is implemented as described in Fletcher (1987).

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