Function Optimization Algorithm Based on Future Search Algorithm

1. Theoretical basis

1. Future Search Algorithms

The Future search algorithm (FSA) is a new search algorithm proposed by M. Elsisi in 2018 by imitating the human yearning for a better life. The algorithm simulates the optimal life between people by establishing a mathematical model (local search) and historical optimal living (global search) to obtain the optimal solution. Compared with other algorithms, FSA has the advantages of less adjustment parameters, fast convergence speed, and strong optimization ability.

(1) Algorithm initialization

FSA initializes the current solution by formula (1): $$S(i,:)=\text{Lb}+(\text{Ub}-\text{Lb}).\ast rand(1,d)\text{\hspace{1em}( 1 )}$$ Among them,$S(i,:)$ means$Current\; solution\; for\; i$ countries;$\text{Ub}$ and$\text{Lb}$ represents the upper and lower bounds of the search space, respectively;$rand$ means$[0,1]$ Interval uniformly distributed random numbers;$d$ represents the problem dimension.

(2) Local solution and global optimal solution

FSA defines the current optimal solution for each country/region as a local optimal solution $\text{LS}$ , which defines the current optimal solution for all countries as the global optimal solution$\text{GS}$ , and obtain the optimal solution of the problem to be optimized through an iterative process. FSA realizes the update of local solution and global optimal solution through equations (2) and (3):$$S(i,:{)}_L=(\text{LS}(i,:)-S(i,:))\ast rand\text{\hspace{1em}(2)}$$$$S(i,:{)}_G=(\text{GS}-S(i,:))\ast rand\text{\hspace{1em}( 3 )}$$ Among them, S (i, :) LS (i,:) _L$S(i,:{)}_L$和$S(i,:{)}_G$respectively represent the $Local\; and\; global\; optimal\; solutions\; for\; i$ countries;$\text{LS}(i,:)$ means$Local\; optimal\; solution\; for\; i$ countries;$\text{GS}$ denotes the global optimal solution for all countries;$rand$ means$[0,1]$ random number in the range.

(3) Define a new solution

in obtaining the $After\; the\; local\; solution\; and\; the\; global\; optimal\; solution\; for\; i$ countries/regions, use Equation (4) to redefine the current solution:$$S(i,:)=S(i,:)+S(i,:{)}_L+S(i,:{)}_G\text{\hspace{1em}(4)}$$

(4) Update the random initial value

FSA is updating the local optimal solution $\text{LS}$ and global optimal solution$\text{After GS}$ , use equation (5) to update the random initial value of equation (1):$$S(i,:)=\text{GS}+(\text{GS}-\text{LS}(i,:))\ast rand\text{\hspace{1em}(5)}$$

2. Flow chart of FSA algorithm

The flow chart of the FSA algorithm is shown in Figure 1.

Figure 1 Flowchart of FSA algorithm

2. Simulation experiment and result analysis

Comparing FSA with PSO and GSA, F1, F2 (unimodal function/30 dimension), F9, F10 (multimodal function/30 dimension), F14, F15 (fixed-dimensional multimodal function/2-dimensional, 4-dimensional) as an example, the experiment set the population size to 30 and the maximum number of iterations to 1000. The results are as follows:

函数：F1
FSA：最优值:0
PSO：最优值:182.2818
GSA：最优值:8.6339e-17
函数：F2
FSA：最优值:0
PSO：最优值:19.0327
GSA：最优值:4.5383e-08
函数：F9
FSA：最优值:0
PSO：最优值:79.9366
GSA：最优值:22.884
函数：F10
FSA：最优值:8.8818e-16
PSO：最优值:6.5874
GSA：最优值:5.8252e-09
函数：F14
FSA：最优值:1.992
PSO：最优值:11.7187
GSA：最优值:2.2116
函数：F15
FSA：最优值:0.00031431
PSO：最优值:0.00030749
GSA：最优值:0.0038895

The experimental results show that the FSA algorithm has better performance.

3. References

[1] M. Elsisi. Future search algorithm for optimization [J]. Evolutionary Intelligence, 2019, 12: 21-31.
[2] Guo Cunwen, Cui Dongwen. PCA-FSA-MLR model and its application in runoff prediction [J] ]. People's Pearl River, 2021, 42(6): 91-98.