The particle swarm optimization technique has two major steps position and velocity update, both equations are defined as follows:

Where the velocity of ith particle is is inertia of particles, and are acceleration coefficients and are random numbers, is personal best of ith particle, is the global best, and X$,$ is the position of ith particle. Determine the followings:

a$)$ Update condition for pbest and gbest

b$)$ Momentum part, Cognitive part, social part of the velocity update formula

c$)$ State the possible three cases