A generalized direct search acceptable-point technique for use with descent-type multivariate algorithms

Many non-linear programming algorithms employ a univariate subprocedure to determine the step length at each multivariate iteration. In recent years much work has been directed toward the development of algorithms which will exhibit favorable convergence properties on well-behaved functions without requiring that the univariate algorithm perform a sequence of one-dimensional minimizations.In this paper a direct search method (the golden section search) is modified to search for acceptable rather than minimizing step lengths and then used as the univariate subprocedure for a generalized conjugate gradient algorithm. The resulting multivariate minimization method is tested on standard unconstrained test functions and a constrained industrial problem. The new method is found to be relatively insensitive to tuning parameters (insofar as success or failure is concerned).A comparison of the golden section acceptable-point search (GSAP) with other popular acceptable-point methods indicates that GSAP is a superior strategy for use with the conjugate directions-type algorithms and is also suitable for use with the quasi-Newton methods. The comparison are based on equivalent function evaluations required to minimize multivariate test functions.