Improving on Golden-Section Optimisation for Locally Symmetric Functions

Abstract

We consider the minimisation of a uniextremal function f(.) on [0,1] using a “second-order” algorithm. At each iteration the current feasible region is resealed to [0,1], so that the optimizing value x* in the initial [0,1]-interval varies from iteration to iteration, which defines a dynamic system. Many line-search algorithms exhibit chaotic behaviour when resealing is applied. If f(.)is symmetric around x*, the associated dynamic system is time-homogeneous and often possesses an invariant density. In a first part, we show that the asymptotic behaviour of the classical Golden-Section algorithm is the same for locally symmetric functions as for pure symmetric functions. We believe that this property is also true for other line-search algorithms, with sometimes a better ergodic rate than the Golden-Section algorithm. In a second part, we consider the case where the number of iterations is fixed a priori, with a dynamic-programming approach, using a uniform prior density on [0,1] for x*.