Investigation of influence of objective function valley ratio on the determination error of its minimum coordinates

Abstract

Objectives. A valley is a region of an objective function landscape in which the function varies along one direction more slowly than along other directions. In order to determine the error of the objective function minimum location in such regions, it is necessary to analyze relations of valley parameters.Methods. A special test function was used in numerical experiments to model valleys with variables across wide ranges of parameters. The position and other valley parameters were defined randomly. Valley dimensionality and ratio were estimated from eigenvalues of the approximated Hessian of objective function in the termination point of minimum search. The error was defined as the Euclidian distance between the known minimum position and the minimum search termination point. Linear regression analysis and approximation with an artificial neural network model were used for statistical processing of experimental data.Results. A linear relation of logarithm of valley ratio to logarithm of minimum position error was obtained. Here, the determination coefficient R2 was ~0.88. By additionally taking into account the Euclidian norm of the objective function gradient in the termination point, R2 can be augmented to ~0.95. However, by using the artificial neural network model, an approximation R2 ~ 0.97 was achieved.Conclusions. The obtained relations may be used for estimating the expected error of extremum coordinates in optimization problems. The described method can be extended to functions having a valley dimensionality of more than one and to other types of hard-to-optimize algorithms regions of objective function landscapes.