Particle Swarm Optimization Algorithm for 1D and 2D Function Optimization

The Particle Swarm Optimization (PSO) algorithm provides an effective approach for solving optimization problems involving 1-dimensional and 2-dimensional objective functions. This method features swarm intelligence principles where particles (potential solutions) navigate the search space by updating their velocity and position based on personal best (pBest) and global best (gBest) values. The algorithm implementation typically includes initialization of particle positions/velocities, fitness evaluation, and iterative updates using velocity equations with inertia weight and acceleration coefficients. This optimization technique can be extensively applied to various multivariate function optimization challenges encountered in engineering design, economic decision-making, and scientific research domains. Through PSO's iterative optimization process, which employs social and cognitive components to balance exploration and exploitation, we can achieve superior results by efficiently locating global optima or near-optimal solutions within the problem space. The algorithm's core functions involve calculating particle velocities (v = w*v + c1*rand()*(pBest-x) + c2*rand()*(gBest-x)) and updating positions (x = x + v) each iteration. Therefore, PSO represents a highly promising optimization methodology capable of addressing diverse complex optimization problems, with particular strength in handling non-convex, multi-modal functions where traditional gradient-based methods may struggle. The algorithm's parallel nature and minimal parameter requirements make it particularly suitable for real-world applications requiring robust optimization performance.