Particle Swarm Optimizer (PSO) Algorithm:

In our Introductory lesson, we learned about the inspiring design principles behind the Particle Swarm Optimizer (PSO) algorithm. In this lesson, we’ll dive deeper into the components that make it perfect for tackling complex optimization problems. Particle Swarm Optimizer (PSO) algorithm uses a swarm of particles, also known as individuals or agents, forming what we call the Population. Each particle in the swarm imitates the successes of its neighbors and itself, creating a simple yet effective behavior. This powerful approach allows the Particle Swarm Optimizer (PSO) algorithm to handle challenging optimization tasks with ease. Let’s explore the fascinating world of PSO and its incredible capabilities together in this lesson!

When particles come together in a swarm. Their collective behavior leads to the exploration and discovery of optimal regions in complex search spaces. Each particle represents a potential solution. To guide their movement, particles rely on a stochastic position update mechanism. This mechanism combines their own knowledge with the knowledge of other particles in the swarm. In essence, particles navigate a multidimensional search space by continuously updating their positions. They learn from their own past experiences and from the experiences of their neighboring particles. This collaborative approach enables them to effectively search for the best solutions. So, in a nutshell, particles work together, learn from each other, and explore the search space to find those optimal spots. Exciting stuff, isn’t it?

Initialization Phase: Particle Swarm Optimizer (PSO) Algorithm

Initialization is the first step in the Particle Swarm Optimizer (PSO) algorithm. We randomly initialize a swarm or population of particles within the bounds of the search space. As particles in the PSO fly through a multi-dimensional search space. That means we have to also initialize the velocity of the initial particles. The following equations provided below allow us to initialize the position and velocity of the particles.

Pop = Lb + (Ub-Lb)*rand(0,1)\\
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Lb: Lower ~Bound ~of ~the~search ~space\\
Ub: Upper ~Bound ~of ~the~search ~space\\
rand(0,1): Uniformly ~distributed ~random~ Numbers \\
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Velocity = Lv + (Uv-Lv)*rand(0,1)\\
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Lv: Lower ~bound~for~the~velocity \\
Uv: Upper ~bound~for~the~velocity \\
rand(0,1): Uniformly ~distributed ~random~ Numbers \\
-------------------------\\  

After initializing the particles with positions and velocities, we assess their fitness by employing the objective function. And, every particle’s fitness is determined based on its current position. We call this fitness value the particle’s “pbest” or personal best. Additionally, we keep track of the best fitness value found by any particle in the swarm. This is known as the “gbest” or global best. This global best represents the optimal fitness value achieved by any particle so far. By continuously updating the pbest and gbest, the algorithm harnesses the power of individual and collective knowledge of the Swarm. Which leads to the discovery of the most promising solutions. It’s through this evaluation and tracking process that the particles make progress toward finding the best possible outcomes.

Position Update Mechanism of PSO algorithm

We update the position of the initial particles using a position update mechanism. And, we also additionally integrate a velocity component to the particle’s position. The velocity component provides momentum for the particles to fly around different regions in the search space. The equation provided below represents the position update mechanism.

X_{i}(t+1) = X_{i}(t) + V_{i}(t+1) \\
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X_{i}(t) ~and~X_{i}(t+1)~ \text{represtents the position of }~i^{th}~\text{particle at time step}~t~ and ~t+1\\
V_{i}(t+1): velocity ~associated~with~i^{th}~particle \\
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You may think of the velocity component as the heart of the Particle Swarm Optimizer (PSO) algorithm. Researchers have proposed various velocity update mechanisms. In this lesson, we will focus on understanding the basic variant of PSO or also known as global PSO. For instance, in global best Particle Swarm Optimization (gbest PSO), each particle’s neighborhood consists of the entire swarm. That means we consider every particle in the swarm as a neighbor to every other particle. The social network employed by gbest PSO reflects the star topology. All particles in a star topology connect to the global best position in the swarm. The global best position represents the best solution found so far by any particle in the entire swarm.

We calculate the velocity of particles in the global best PSO using the following equation;

V_{i}(t+1) = V_{i}(t) + c_{1}r_{1}(X_{pbest}-X_{i}) + c_{2}r_{2}(X_{gbest} - X_{i})\\
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V_{i}(t): Velocity ~of ~i^{th}~ particle~at ~time~t\\
X_{i}: position~of~current~particle~i\\
X_{pbest}: particle's~best~position\\
X_{gbest}: global~best~position~in~the~swarm\\
c_{1}: cognitive~ component\\
c_{2}: social~component \\
r_{1}, r_{2}: random~numbers \\
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As you can observe from the above equation, c1 is the cognitive component and c2 is the social component. The social and cognitive components decide the new position of the particle in the search space. First, Let’s understand the social and cognitive components in detail.

Social and Cognitive Knowledge of Particle Swarm:

The social component in the Particle Swarm Optimizer (PSO) represents the influence of neighboring particles on an individual particle’s movement. It captures the idea of collaboration and information sharing within the swarm. Each particle adjusts its velocity based on the best solution found by its neighboring particles. The social component allows a particle to explore and exploit the search space by considering the successes of other particles. It promotes cooperation among the particles in finding better solutions.

And, the cognitive component in the Particle Swarm Optimizer (PSO) algorithm represents the particle’s individual knowledge and experience. It reflects the particle’s own best solution found so far, known as the “personal best” or “local best” position. The cognitive component allows a particle to remember its own historical best position. The cognitive component provides a sense of memory to each particle. This enables PSO to retain information about the best solution it has encountered. It helps particles to remember promising regions they have explored in the search space. This means, by considering their personal best position, particles can exploit their own knowledge and experiences, making adjustments to their movement accordingly.

So, in short, we may say, the social and cognitive components work together to update the particle’s velocity. And, this influences its movement in the search space. We achieve a balance between these two components by combining the parameter values in a weighted manner. And, these coefficients determine the relative importance of the social and cognitive components in influencing the particle’s behavior. That means, incorporating both social and cognitive components. The Particle Swam Optimization (PSO) combines exploration and exploitation to efficiently search the problem space.

A Greedy Selection Mechanism

Now comes the important principle of the selection mechanism. Immediately after updating the positions of the particles in the search space. We evaluate the fitness value using the underlying objective function. For selecting the better particle, many selection mechanisms like Greedy Selection, and Tournament Selection exist. However, the greedy selection mechanism is the most used strategy for selection. In greedy selection, we compare the previous fitness values with the current fitness values. And, then we select the particle with the minimum fitness. This is how we update the position of particles in the search space. The process is iteratively continued for all the particles until the termination criteria are satisfied.

Applications of Particle Swarm Optimizer (PSO) algorithm:

PSO algorithm is one of the first popular meta-heuristic algorithms under the class of swarm-inspired algorithms. Researchers have developed a new variety of algorithms in the recent past, drawing inspiration from the PSO algorithm. And, here are some of the popular use cases of the PSO algorithm in artificial intelligence and machine learning.

We have included just a few examples of how Particle Swarm Optimizer PSO can be applied to AI and ML problems. The flexibility and adaptability of the PSO algorithm make it a valuable tool for solving a wide range of problems. Why not implement MATLAB or Python code for the Particle Swarm Optimization Algorithm in your favorite computing environment. Or if you want to read the original paper of PSO published by Kennedy and Ebherhert, click here.

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