On optimization methods in the problems of experiment design

Abstract

A new known modification for simulation of annealing to search the global extremum of the functions of many variables uses the fact that the function when n → ¥ converges to the δ-function concentrated at the point of global maximum of f(x). The case when the function has many equal extrema is discussed in detail. Problems of this type are often present, particularly in the design of regression experiments. Here we introduce the reader to an extremum search method that is effective in solving a wide range of applied problems, and also illustrate the use of the method in some of the simplest problems of designing the regression experiments. The proposed modification of simulated annealing uses quasi-random search at the intermediate stages. This is not the most rapid, but very reliable method which provide a complete exploring of the function domain. When solving numerical examples, the so-called exact D-optimal designs are constructed, which are very difficult to be obtained by other methods. Although with the increase in the number of variables, the complexity of the method (as well as the complexity of other well-known methods) increases dramatically due to an increase in the order of the determinant, the proposed algorithm is simple, reliable, and easily parallelized. It is known that the gain from using optimal designs in some cases can justify any computational costs of developing those designs. Using the described technique, the reader will be able to construct (even using the laptop capacity) the optimal designs in different areas at moderate values of the parameters (for example, for quadratic regression for s variables in variables for s ≤ 10).