Convergence Analysis of Particle Swarm Optimizer and Its Improved Algorithm Based on Velocity Differential Evolution

Algorithm 1

Procedure for the H-PSO-DE.

Step 1. Initialize particles in swarm. Initialize location 𝑥_𝑛, and velocity 𝑣_𝑛

   of each particle 𝑛. Best position 𝑦_𝑛=𝑥_𝑛.

Step 2. Evaluate objective function 𝑓(𝑥_𝑛) and update personal best.

   for each particle 𝑛, do

      if 𝑓(𝑥_𝑛)<𝑓(𝑦_𝑛) then

     𝑦_𝑛=𝑥_𝑛

    end if

   end for

Step 3. Swap particles.

     for each particle 𝑛, do

     Determine the best successor 𝑚= arg

     min 𝑓𝑦_𝑞∣𝑞∈𝑆(𝑛), where are the successors of 𝑛.

    if 𝑓(𝑦_𝑚)<𝑓(𝑦_𝑛) then

     Swap particles 𝑛 and 𝑚

    end if

   end for

Step 4. Update the position and velocity of the 𝑛th particle.

   for each particle 𝑛, do

  Update the velocity 𝑣_𝑛 in each dimension 𝑑:

    𝑣_𝑛,𝑑=𝑤𝑣_𝑛,𝑑+𝑐₁𝑟₁𝑦_𝑛,𝑑−𝑥_𝑛,𝑑+𝑐₂𝑟₂𝑦_{𝑝(𝑛),𝑑}−𝑥_𝑛,𝑑

    Move the particle: 𝑥_𝑛,𝑑=𝑥_𝑛,𝑑+𝑣_𝑛,𝑑

   end for

Step 5. Judge the evolution process of H-PSO.

   if (𝑔₀==𝐺₀), then

    goto Step 6

    else goto Step 7

   end if

Step 6. Update the velocity and position of the particle according to (16).

   if 𝑓(𝑥^′_𝑛)<𝑓(𝑥_𝑛) then 𝑥_𝑛=𝑥^′_𝑛

   end if

Step 7. If a stopping criterion is met, then output the global best position and stop;

   otherwise, repeat Step 2–Step 6.