Mixed Methods of Optimisation

Mixed methods of optimisation are combination of random methods and deter­ministic methods. Below, we will discuss one of them, the complex method. This method assumes that in an N-dimensional area R, at least one starting point is known. We will divide the restrictions of the R area into two groups:

• functional restrictions:

Ui(xn, P < °) i = 1; •••, m,

• known restrictions:

In the first stage, to a known starting point, we sample more, until the total number of points amounts to k > n + 1. The sampling is carried out in the area of known restrictions, and then, we check whether the sampled point meets the functional restrictions. If yes, the point is considered as favourable, and if not, the sampled point is moved along the straight line passing through the sampled point and the last favourable point in the direction of a favourable point for half the length of the section between them. This procedure is repeated as long as certain condi­tions are met. Figure 6.107 illustrates this procedure. Let us assume that as a result of the sampling, we obtain subsequently the points xj, x0, i x3. The first two points are favourable and the third not; therefore, we move it along the straight line passing through the points x°, i x° in the direction of the point x° by half the distance between points x0, i x3. This way, we obtain the point x4. In this example, already one move was enough, because the point x4 turned out to be favourable. A set of points obtained in this way has been named “a complex” by the author of the method. In the discussed example, three starting points are enough, because the area of decision variables is two-dimensional.

In the second and subsequent stages, the procedure is as follows: the centre of gravity of x03 points of the complex is determined, as well as the value of the function of purpose in each of these points. The most unfavourable point is selected and replaced with another point, whereas the coefficient of homotheticity a should be greater than 1. If a new point does not depend on the area R, then it is moved in the direction of the centre of gravity by half the length of the section between them, this procedure is carried out as long as the new point is found in the area R. After completing this procedure, a new complex is obtained, on which identical opera­tions as on the starting complex are made. In the example shown in Fig. 6.107, the new complex is points xj, x4, i x^. The criterion to stop searching for the optimum in this method is to achieve the equality of the function of purpose, at all points in the complex, with the precision set by the user.