On the Convergence of the P-Algorithm for One-Dimensional Global Optimization of Smooth Functions

A Global Optimization Method Based on the Reduced Simplicial Statistical Model

“…Then for the construction of the algorithm only the statistical model defined over edges of simplices is needed. Based on the comparison of various statistical models in [11] and on our past experience, we chose the statistical smooth function model defined in [1] to model the behaviour of the objective function over the edges of the simplicial partition. That model is a stationary Gaussian stochastic process with the correlation function r(·) satisfying the following regularity assumptions…”

“…Assume that the next point for computation of f (x) can be chosen on an edge only, and that the behaviour of f (x) over an edge is modeled by the smooth function model. The latter assumption implies that the maximum point of max x P ij k (x) = P{ξ(x) ≤ y ok } where x belongs to the segment of line connecting vertices x i and x j can be computed by a simple formula [1]. An appropriate monotonically increasing function of max x P ij k (x) can be used to compute the criterion for the selection of simplices.…”

“…An example of that sub-class is the Wiener process which was the first model used for those purposes [10]. However, for smooth functions the P-algorithm based on the smooth function model has better convergence properties as shown in [1]. Our idea is to apply the efficient onedimensional P-algorithm based on the smooth function model as a subroutine of a multidimensional search algorithm.…”

P-algorithm based on a simplicial statistical model of multimodal functions

“…This property is especially favorable since it implies unimodality of u k (x) over the sub-interval (x k i−1 , x k i ); moreover, the maximum of u k (x) can be defined by an analytical formula involving only the information attributed to the sub-interval. It is shown in Calvin and Žilinskas (1999) that the maximum point of u k (x) (i.e., the next point for computing of value of the objective function f (·)) can be found using the formula…”

Univariate Global Optimization with Multiextremal Non-Differentiable Constraints Without Penalty Functions

“…In last decades univariate global optimization problems were studied intensively (see [7,15,18,19,22,28,31,37,43]) because there exists a large number of real-life applications where it is necessary to solve such problems (see [6,15,27,30,33,37,40]). On the other hand, it is important to study these problems because mathematical approaches developed to solve them can be generalized to the multidimensional case by numerous schemes (see, for example, one-point based, diagonal, simplicial, spacefilling curves, and other popular approaches in [14,16,17,23,26,29,37]).…”