the dimension of the benchmark functions. With the increase of dimension of the functions, the size of the population of the proposed SPBO needs not to be increased. That’s why the population size of SPBO for all the considered benchmark functions is considered to be constant. It is considered as 20 for the proposed SPBO. In order to have a fair comparison of the performance of all the algorithms, the analysis is done based on the number of fitness function evaluations (NFFE) taken to converge.
Figure 1.1 Flowchart of the SPBO algorithm.
Table 1.1 CEC 2005 benchmark function [67].
| Problem | Type of the function | Name of the functions | F(x*) | Initial range | Bounds | Dimension (D) |
|---|---|---|---|---|---|---|
| F1 | Unimodal | Shifted Sphere Function | -450 | [-100,100]D | [-100,100] D | 30 |
| F2 | Unimodal | Shifted Schwefel’s Problem 1.2 | -450 | [-100,100] D | [-100,100] D | 30 |
| F3 | Unimodal | Shifted Rotated High Conditioned Elliptic Function | -450 | [-100,100] D | [-100,100] D | 30 |
| F4 | Unimodal | Shifted Schwefel’s Problem 1.2 with Noise in Fitness | -450 | [-100,100] D | [-100,100] D | 30 |
| F5 | Unimodal | Schwefel’s Problems 2.6 with Global Optimum on Bounds | -310 | [-100,100] D | [-100, 100] D | 30 |
| F6 | Basic multimodal | Shifted Rosenbrock’s Function | 390 | [-100, 100] D | [-100, 100] D | 30 |
| F7 | Basic multimodal | Shifted Rotated Griewank’s Function without Bounds | -180 | [0, 600] D | [0, 600] D | 30 |
| F8 | Basic multimodal | Shifted Rotated Ackley’s Function with Global Optimum on Bounds | -140 | [-32, 32] D | [-32, 32] D | 30 |
| F9 | Basic multimodal | Shifted Rastrigin’s Function | -330 | [-5, 5] D | [-5, 5] D | 30 |
| F10 | Basic multimodal | Shifted Rotated Rastrigin’s Function | -330 | [-5, 5] D | [-5, 5] D | 30 |
1.2.3 Mixed Discrete SPBO
In general, the optimization algorithm is used to optimize (minimize) the objective function by obtaining the optimum value of the variable vector X. To optimize the variable vector X, consist of some continuous and discrete variables, a mixed discrete version of SPBO may be used. For an n-dimensional problem that includes continuous and discrete variables, the variable vector may be represented as in (Equation 1.5).
where [Xcont] and [Xdisc] are the continuous and discrete variable vectors, respectively. For the scenario of m continuous variables and remaining (n-m) discrete variables, the [Xcont] and [Xdisc] maybe expressed as in (Equation 1.6) and (Equation 1.7), respectively.
As mentioned earlier the mixed discrete SPBO is capable to handle both the continuous and discrete variables. In the mixed discrete SPBO, the continuous variables are updated as the conventional SPBO. The updating process of the continuous variables is the same as discussed in the previous section using four categories of students (namely best student, good student, average student, and students who want to improve randomly) with the help of (Equations 1.1–1.4). For the discrete variables, the discretization may be done with the help of the nearest vertex approach (NVA). The NVA method is normally based on finding out the Euclidean norm in the design space. The discrete variables may be expressed in terms of a hypercube, which are represented by the sets of ordered pair and can be represented as in (Equation 1.8)
where,