An Approach to Multidimensional Nonlinear Optimization

Abstract

The article presents a new approach to improving the efficiency of multidimensional nonlinear optimization. Traditional linear search is replaced by non-linear one, in which the direction of search at each step adapts to the profile of the objective function. This makes it possible to localize the extremum as quickly as possible and to shorten substantially the time of its determination. An interpolation search algorithm is proposed in the found interval of extremum localization. The objective function is modeled by a segment of a cubic spline, constructed on the basis of information about the gradient vector at the boundary points and taking into account the slope of the objective function. This allows to reduce the number of stages of interpolation search. The possibility of simplified nonsmooth interpolation by first-order splines in the area of finding an extremum is considered. Numerical results show that the new method is very efficient for the various test problems.