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Abstract
Immune cloud particle swarm optimization (ICPSO) is a novel optimization algorithm based on particle swarm optimization (PSO) and immune algorithm. The ICPSO algorithm is designed to solve the multi-modal optimization problems. The algorithm integrates the cloud model theory and immune mechanism into PSO operations. The cloud model theory is used to model the uncertainty of the search space, and the immune mechanism ensures the diversity of the particle swarm. In this paper, we will introduce the basic idea of ICPSO, its advantages, and the comparison with other optimization algorithms. Finally, we will demonstrate the feasibility of the proposed method through simulation experiments.
Introduction
Multi-modal optimization problems are common in the fields of engineering, science, economics, and so on. The multi-modal optimization problems refer to the optimization problems that have multiple optimal solutions rather than a single optimal solution. The traditional optimization algorithms, such as gradient-based methods, random search, and genetic algorithms, are not suitable for the multi-modal optimization problems. Because these algorithms will converge to a local optimum, which may not be the global optimum. To overcome this problem, researchers have proposed many optimization algorithms, such as particle swarm optimization, differential evolution, and ant colony optimization. Among these algorithms, particle swarm optimization (PSO) has attracted much attention for its simplicity and efficiency.
Particle swarm optimization is a population-based optimization algorithm that simulates the swarm behavior of birds or fish. PSO has been successfully applied to single-modal optimization problems. However, it may encounter difficulties in solving multi-modal optimization problems, where the search space has multiple optimal solutions. PSO can easily fall into local optima due to the lack of diversity in the particle swarm.
To address this issue, researchers have proposed many modified versions of PSO, such as Cooperative PSO (CPSO), Multi-swarm PSO (MSPSO), and Quantum-behaved PSO (QPSO). These algorithms introduce various mechanisms to enhance the diversity of particles. However, they often increase the computational cost or the complexity of the algorithm.
Immune algorithm is another type of optimization algorithm that mimics the immune system of the human body. Immune algorithms have been applied to various optimization problems and have shown good performance in solving complex optimization problems. The basic mechanism of immune algorithm is to maintain diversity and detect similar solutions.
Recently, some researchers have proposed an immune-based particle swarm optimization algorithm, which is called the immune particle swarm optimization (IPSO). IPSO combines the swarm intelligence of PSO and the immune mechanism. However, the IPSO algorithm still has some limitations in solving multi-modal optimization problems.
To address the limitations of PSO and IPSO in solving multi-modal optimization problems, some researchers have proposed an immune cloud particle swarm optimization (ICPSO) algorithm. The ICPSO algorithm is an optimization algorithm based on PSO, immune algorithm and cloud model theory. The ICPSO algorithm can model the uncertainty of the search space using the cloud model theory, and can maintain the diversity of the particle swarm using the immune mechanism. The algorithm has achieved good performance in solving multi-modal optimization problems.
The remainder of this paper is organized as follows. In Section II, we introduce the basic idea of ICPSO. In Section III, we discuss the advantages of ICPSO over other optimization algorithms. In Section IV, we present the simulation experiments to demonstrate the feasibility of the proposed method. Finally, in Section V, we conclude the paper and provide some insights for future research.
II. The Basic Idea of ICPSO
In this section, we introduce the basic idea of ICPSO algorithm. The ICPSO algorithm integrates the PSO, immune algorithm and cloud model theory. The cloud model theory is used to model the uncertainty in the search space. The immune algorithm is used to ensure the diversity of the particle swarm.
A. Particle Swarm Optimization
The particle swarm optimization is a population-based optimization algorithm that mimics the swarm behavior of birds or fish. In PSO, each particle represents a potential solution to the optimization problem. The fitness value of each particle is evaluated based on the objective function. The particles move in the search space according to their current position and the experience of the particle swarm. The velocity of each particle is updated by three factors: the cognitive factor, the social factor, and the random factor. The cognitive factor represents the best position that the particle itself has reached. The social factor represents the best position that has been reached by the particle swarm. The random factor is used to introduce stochasticity into the algorithm. The PSO algorithm can be summarized as follows:
1. Initialize the particle swarm randomly.
2. Evaluate the fitness of each particle.
3. Initialize the velocity of each particle randomly.
4. Update the position of each particle using Eq. (1).
x_(t+1,i)=x_t,i+v_(t,i) (1)
where x_t,i is the position of the ith particle at iteration t, v_(t,i) is the velocity of the ith particle at iteration t.
5. Update the velocity of each particle using Eq. (2).
v_(t+1,i)=ωv_t,i+c_(1)r_(1)(p_best,i-x_(t,i))+c_(2)r_(2)(g_best-x_(t,i)) (2)
where ω is the inertia weight, c_(1) and c_(2) are the learning factors, r_(1) and r_(2) are the random values between 0 and 1, p_best,i is the best position that the ith particle has reached, and g_best is the best position that has been reached by the particle swarm.
6. Evaluate the fitness of each updated particle.
7. Update the p_best and g_best of each particle.
8. Repeat Steps 4 to 7 until the stopping criterion is met.
B. Cloud Model Theory
The cloud model theory is a new method for modeling the uncertainty in the search space. The cloud model theory is based on the idea that the probability density function (PDF) and the cumulative distribution function (CDF) are two complementary descriptions of random events. The cloud model theory can model the randomness, fuzziness and uncertainty in the optimization problem.
The cloud model can be defined as a triple (α,β,γ), where α is the expectation, β is the deviation, and γ is the randomness. The expectation α and the deviation β can be computed from the mean and standard deviation of the data. The randomness γ can be computed from the skewness and kurtosis of the data.
The probability density function (PDF) of the cloud model is given by:
p(x)=[1/(β(Γ(1/γ)))][1+(γ/β)(x-α)]^(-1/γ-1) (3)
where Γ(·) is the gamma function.
The cumulative distribution function (CDF) of the cloud model is given by:
F(x)=[1/(1+γ(x-α)/β))^γ] (4)
The cloud model can be used to generate random numbers that conform to the distribution of the data. The cloud model theory can also be used to simulate the uncertainty of the search space.
C. Immune Mechanism
The immune mechanism is used to ensure the diversity of the particle swarm. The immune mechanism is based on the idea that the immune system of the human body can detect and eliminate the invading pathogens. The immune mechanism can be divided into three stages: selection, cloning, and mutation.
In the selection stage, the particles are selected based on their fitness. The fitter particles have a higher probability of survival.
In the cloning stage, the selected particles are cloned to produce a new set of particles. The number of clones is proportional to the fitness of the particles.
In the mutation stage, the cloned particles are mutated to introduce diversity into the particle swarm. The mutation operation randomly changes the position of the cloned particles within a certain range.
The immune mechanism can be used to enhance the diversity of the particle swarm and prevent the particles from converging to a local optimum.
D. Immune Cloud Particle Swarm Optimization
The immune cloud particle swarm optimization integrates the PSO, the cloud model theory, and the immune mechanism. The ICPSO algorithm can be summarized as follows:
1. Initialize the particle swarm randomly.
2. Evaluate the fitness of each particle.
3. Initialize the cloud models for each particle.
4. Initialize the velocity of each particle randomly.
5. Update the position of each particle using Eq. (5).
x_(t+1,i)=x_t,i+v_(t,i) (5)
where x_t,i is the position of the ith particle at iteration t, v_(t,i) is the velocity of the ith particle at iteration t.
6. Update the velocity of each particle using Eq. (6).
v_(t+1,i)=ωv_t,i+c_(1)r_(1)(p_best,i-x_(t,i))+c_(2)r_(2)(g_best-x_(t,i)) (6)
where ω is the inertia weight, c_(1) and c_(2) are the learning factors, r_(1) and r_(2) are the random values between 0 and 1, p_best,i is the best position that the ith particle has reached, and g_best is the best position that has been reached by the particle swarm.
7. Evaluate the fitness of each updated particle.
8. Update the p_best and g_best of each particle.
9. Update the cloud models of each particle using the Eq. (7) to (9).
α_(t+1,i)=α_t,i+θα[(g_best-x_(t,i))/d]+(1-θα)φ_αxi (7)
β_(t+1,i)=β_t,i+θβ[|y(p_best,i)-yp|i|/d]+(1-θβ)φ_β|y(xi)-yp|i| (8)
γ_(t+1,i)=γ_t,i+θγ[(f_best-f(xi))/d]+(1-θγ)φ_γfi (9)
where θα, θβ, and θγ are the weight coefficients, d is the maximum distance between the particles, φ_αxi, φ_β|y(xi)-yp|i|, and φ_γfi are the random values generated from the cloud models of the particles, yp|i is the average fitness of the particles in the neighborhood of xi, and f(xi) is the fitness of the particle xi.
10. Apply the immune mechanism on the top k particles to ensure the diversity of the particle swarm.
11. Repeat Steps 5 to 10 until the stopping criterion is met.
III. Advantages of ICPSO
The ICPSO algorithm has several advantages over other optimization algorithms:
A. Modeling the Uncertainty
The cloud model theory in the ICPSO algorithm can model the uncertainty of the search space. The cloud model theory can take into account the randomness, fuzziness, and uncertainty of the optimization problem. The cloud model theory can also generate random numbers that conform to the distribution of the data. The ICPSO algorithm can deal with the multi-modal optimization problems that have multiple optimal solutions.
B. Ensuring the Diversity
The immune mechanism in the ICPSO algorithm can ensure the diversity of the particle swarm. The immune mechanism can prevent the particle swarm from converging to a local optimum. The immune mechanism can also detect and eliminate similar solutions in the particle swarm. The ICPSO algorithm can maintain a good balance between exploration and exploitation.
C. High Efficiency
The ICPSO algorithm is a simple and efficient optimization algorithm. The ICPSO algorithm requires few parameters to be tuned. The ICPSO algorithm can converge quickly and obtain good solutions in a short time. The ICPSO algorithm can be easily implemented and scaled up to large-scale optimization problems.
IV. Simulation Experiments
In this section, we present simulation experiments to demonstrate the feasibility of the proposed method. We test the ICPSO algorithm on six benchmark optimization problems. The six problems are: Rastrigin, Ackley, Griewank, Rosenbrock, Schwefel, and Weierstrass. The dimensions of the problems range from 2 to 100.
We compare the performance of ICPSO with that of other optimization algorithms, including standard PSO, CPSO, and IPSO. We also compare the per