Fairly Constricted Particle Swarm Optimization

We have adapted the use of exponentially averaged momentum in PSO to multi-objective optimization problems. The algorithm was built on top of SMPSO, a state-of-the-art MOO solver, and we present a novel mathematical analysis of constriction fairness. We extend this analysis to the use of momentum and propose rich alternatives of parameter sets which are theoretically sound. We call our proposed algorithm "Fairly Constricted PSO with Exponentially-Averaged Momentum" — FCPSO-em 2 . I. PARTICLE SWARM OPTIMIZATION A. Vanilla PSO Particle Swarm Optimization (PSO) was first proposed by Kennedy and Eberhart [1], [2] in 1995 as an attempt to model the behaviour of bird flocks. N particles are initialised at random positions/velocities in the search space, ith particle updates its trajectory according to — v (t+1) i = wv (t) i + c1r1(pbest (t) i − x (t) i ) + c2r2(gbest (t) − x i ) (1) x (t+1) i = x (t) i + v (t+1) i (2) r1 and r2 are D-dimensional random vectors with each component in U(0, 1). pbest i is the best position that particle i has visited upto time t i.e. achieved the lowest value in the objective function f(x). gbest is the best position among all particles that has been achieved. B. Exponentially-Averaged Momentum One technique to avoid premature convergence in PSO is to introduce an exponentially-averaged momentum (EM) term that tracks the histories of a particle’s velocities. Hence, if a particle is near a local optima, the influence of momentum will push it out and it will subsequently contribute to the global search. With the momentum term as M (t) i — M (t+1) i = βM (t) i + (1− β)v (t) i (3) v (t+1) i = M (t+1) i + c1r1(pbest (t) i − x (t) i ) + c2r2(gbest (t) − x i ) (4) The position update equation remains the same as eq (2). The factor β controls the preference of momentum over velocity. 1Corresponding Author Snehanshu Saha 2GitHub Repository https://github.com/anuwu/jMetalPy 3Anonymous Google Drive with Results link β = 0 degenerates to vanilla PSO. By recursively expanding eq (3), a particle’s momentum is an exponentially weighted sum of all its previous velocities — M (t+1) i = (1− β)v (t) i + β(1− β)v (t−1) i + . . . + βt−2(1− β)v i + β t−1(1− β)v i (5) Such a PSO algorithm has been devised in [3] and they have reported significant reduction in the number of iterations for single-objective problems (upto 50%), and avoidance of premature convergence. Definition 1 A PSO system that uses EM is called an EMPSO algorithm.