Particle swarms can easily be used to optimize a function with a set of linear equality constraints, by restricting the swarm’s movement to the constrained search space. A “Linear Particle Swarm Optimiser” and “Converging Linear Particle Swarm Optimiser” is developed to optimize linear equality-constrained functions. It is shown that if the entire swarm of particles is initialized to consist of only feasible solutions, then the swarm can optimize the constrained objective function without ever again considering the set of constraints. The Converging Linear Particle Swarm Optimiser overcomes the Linear Particle Swarm Optimiser’s possibility of premature convergence. Training a Support Vector Machine requires solving a constrained quadratic programming problem, and the Converging Linear Particle Swarm Optimiser ideally fits the needs of an optimization method for Support Vector Machine training. Particle swarms are intuitive and easy to implement, and is presented as an alternative to current numeric Support Vector Machine training methods.