Numerical Investigation of Some Potential Problems of Univariate Minimization Methods

Abstract

Many nonlinear programming algorithms employ a univariate subprocedure to determine the step length at each multivariate iteration. In this note, a popular polynomial approximation-interpolation univariate algorithm (DSC-P) is compared to two versions of the golden section search. One-dimensional test functions which model the behavior of barrier and penalty functions are used for the comparison. In general, the polynominal method indicates convergence in fewer function evaluations than the golden section search. However, it is significantly less reliable. Tight convergence criteria do not necessarily lead to accurate results with the polynomial-based univariate strategy. A companion paper provides a theoretical basis for the observations and gives conditions underwhich DSC-P will fail—even for strictly convex, unimodal functions and exact arithmetic.