Physics
Computer Science-2021.gz:143175
Computer Science
1830e8468fc62af4bdfb250ea457b31d46a604b5
Dynamic Optimization Algorithm Based on Multi-modal Niche Lion Swarm by Sensitive Individuals

Aiming at the problem that the model of complex dynamic system which is difficult to track its global optimal solution, this paper proposes a multi-modal niche lion swarm optimization algorithm based on sensitive individuals to solve the optimization problem of complex dynamic system. Sensitive individuals are set in the lion swarm algorithm to perceive changes in the dynamic environment, and to track the global optimal value in time. It combines with niche technology to increase the diversity of the population and avoids falling into local extremes. The dynamic environment is simulated by the dynamic function generator and compared with other optimization algorithm to verify the superiority of the algorithm proposed in this paper.


Introduction
In recent years, in the fields of production scheduling, artificial intelligence, combination optimization, engineering design, large-scale data processing, network communications, data mining, and capital budgeting, many complex dynamics that are closer to real life are often encountered relative to static optimization problems. Dynamic optimization problem (DOP) [1][2][3][4] means that its objective function is not only related to decision variables, but also changes with time (environment), so its optimal solution will also change with time (environment), Static optimization problem (Static optimization problem, SOP) means that its objective function is only related to decision variables, and its optimal solution does not change with time (environment). In recent years, DOP has attracted more and more researchers' interest, and becomes a research hotspot in computational intelligence.
In order to track the best solution after environmental changes, Blackwell and Bentley [5] studied the use of charged particles in a dynamic environment. The use of charged particles could prevent the clusters of particles from converging into a single point. However, the consequence of this was that the particles would increase the repulsive force and increase the computational cost; Hu and Eberthart proposed a method to monitor the global optimal solution and the sub-global optimal solution to detect changes in the optimal solution [6]; Ursem [7] demonstrated a multi-population method in which the grouping of sub-populations was determined by the "peak-valley detection process", but the peakvalley detection method only worked on points between known optimal values. The particles in the remaining space still could not be sampled.
The intelligent optimization algorithm provides a general framework for solving optimization problems of complex systems. It does not depend on the specific field of the problem, and does not require a clear analytical expression of the objective function, it has strong robustness to the type of problem, so intelligent Optimization algorithms have received more and more attention, in addition to IOP Publishing doi:10.1088/1742-6596/2025/1/012048 2 the traditional classic intelligent optimization algorithms, such as particle swarm optimization (PSO) [8], Artificial Bee Colony Algorithm (ABC) [9], Ant Colony Optimization Algorithm (ACO) [10] and many more. The lion swarm optimization algorithm [11] is a new intelligent optimization algorithm proposed in 2018. Compared with the traditional optimization algorithm, this algorithm has many advantages. It uses the lion swarm to be divided into the lion king, the lioness and the cub and simulates three intelligent behaviors, the lion king to guard the territory, lioness cooperative hunting, cubs follow the lion king and lioness. It can be seen that the algorithm has relatively good co-evolution capabilities, and the communication mechanism is more complex and diversified, which provides us with a new idea for solving complex optimization problems. The LSO itself is a static optimization algorithm. At present, the research and improvement of this algorithm are mostly carried out in the field of static optimization, while the field of dynamic environment optimization is rarely involved. Therefore, solving dynamic optimization problems by using the parallelism of the lions optimization algorithm, the effectiveness and practicability of the global optimization problem is a new method of solving complex optimization problems.

Dynamic Optimization Problem
Compared with static environment problems, dynamic environment optimization problems [12] not only need to be able to track the optimal solution, but also to track the change trajectory of the optimal solution as much as possible. The dynamic optimization problem is generally described as:

Dynamic Environment Creation
Angeline [13] proposed a method to create a series of dynamic environments based on the threedimensional parabolic function of the following formula: The Angeline function can generate three different dynamic environments: linear, circular and random trajectory dynamic environments. The use of this parabolic function method is limited to simple dynamic environments.
However, practical problems are extremely complex, highly non-linear and multi-peak dynamic environments. Morrison and De Jong proposed the dynamic environment test function generator DF1 [14] function. It can specify a certain number of cones or peaks to generate a complex environment within a specified number. For two-dimensional problems, the static evaluation function in the DF1 function is defined as: In equation (6), A is a constant and t Y is the value at the t iteration. A series of different Y values can be generated by changing the A value.

Original Lion Swarm Optimization Algorithm (LSO)
LSO is an abstract organization form based on the cooperative hunting behavior of lion swarm. The location of king is updated according to: where g k is the optimal position of the k th generation population, γ is a random number subject to distribution. The position of the hunter is updated according to: where is the historical optimal position of the i th lion and the k th generation, is the historical optimal position for cubs to follow the hunter ℎ in the generation, ̅ is the position where the ℎ cub is driven in the hunting range.  The fitness fluctuation function of the individual in the sequence is defined as follows:

Improved Lion Swarm
indicates the fitness of the individual in the sequence, max F and min F are the maximum and minimum values of the fitness function in the population, respectively. When the fluctuation is less than a preset threshold, the fitness increment between two adjacent individuals can be ignored.

Lion Swarm Optimization Algorithm
Based on Sensitive Particles. The dynamic particle swarm algorithm based on sensitive particles [15] is a typical dynamic particle swarm algorithm. It randomly selects one or more positions when the algorithm is initialized, called sensitive particles, and calculates the fitness value of sensitive particles in each iteration. When the fitness value changes, it is considered that the environment has changed. The way to respond is to reinitialize the particle position and particle velocity according to a certain proportion. Similar to the introduction of sensitive particles into the PSO algorithm to perceive changes in the dynamic environment, and the introduction of sensitive lions into the lion optimization algorithm, the sensitive lions are re-initialized after they perceive environmental changes, so that the lions can follow the changes in the target position as a whole.

Dynamic Optimization Algorithm Based on Multi-modal Niche Lion Swarm by Sensitive Individuals (SDLSO)
Multi-modal optimization in a dynamic environment using lion optimization algorithm. It is divided into two steps. The first step is to detect changes in the environment and evaluate the changed environment, the second step is to apply the lion optimization algorithm to the detected current environment and combine the niche technology to perform multi-modal optimization.

Experimental Parameter Settings
The experiment uses MATLAB R2019b to simulate. For the SDLSO, set the population size N=20, and the upper limit of the number of iterations is T=800. The function dimension is D=2, and a single experiment is run 30 times, each runs in order to ensure fairness, the initial position of the algorithm is the same as the global optimal solution and the initial positions of the particles are generated randomly. The initial height A of all vertebral bodies is evenly distributed within the allowable range. In order to effectively verify the experimental effect of the SDLSO algorithm in a dynamic environment, this experiment uses particle swarm optimization (PSO) as a comparison. Except that the above parameter settings are consistent, the two learning factors of the PSO algorithm are both 1.49, and w =0.7.

Experimental Results
The experimental results of the PSO algorithm and the proposed SDLSO algorithm are shown in figure 2. The optimization result shows that when the number of dynamic function changes is i=1:120, the optimal value is h=450-fix(i/5); When i=121: 790, h=430; when i=791:1000, h=350+fix(i-500/10) *3; when i=1001:1200, h=500. A single experiment is run 30 times, and the average value is finally obtained. From the curve, we can see that compared with PSO, the improved SDLSO has a better fitness value, and they can follow up in time at the turning points of the curve. The turning points indicate changes in the external environment. Experimental results show that the SDLSO algorithm can successfully track the optimal solution in a two-dimensional dynamic environment.

Conclusion
Aiming at the of a complex dynamic system changing with time and being difficult to track its global optimal solution, this paper proposes a multi-modal niche lion swarm optimization algorithm based on sensitive individuals for dynamic problem. It has ability to perceive changes in the external environment, track changes in the environment in time, and solve complex dynamic system optimization problems. And the introduction of niche technology has improved the diversity of the population and avoided falling into local extremes. It uses the dynamic generator to build the environment, and also compares the effectiveness of other dynamic optimization algorithm with that of the proposed algorithm.