Minimization of Keane’s Bump Function by the Repulsive Particle Swarm and the Differential Evolution Methods

Keane’s bump function is considered as a standard benchmark for nonlinear constrained optimization. It is highly multi-modal and its optimum is located at the non-linear constrained boundary. The true minimum of this function is, perhaps, unknown. We intend in this paper to optimize Keane’s function of different dimensions (2 to 100) by the Repulsive Particle Swarm and Differential Evolution methods. The DE optimization program has gone a long way to obtain the optimum results. However, the Repulsive Particle Swarm optimization has faltered. We have also conjectured that the values of the decision variables diminish with the increasing index values and they form two distinct clusters with almost equal number of members. These regularities indicate whether the function could attain a minimum or (at least) has reached close to the minimum. We have used this conjecture to incorporate ordering of variable values before evalution of the function and its optimization at every trial. As a result, the performance of DE as well as the RPS has improved significantly. Our results are comparable with the best results available in the literature on optimization of Keane function. Our two findings are notable: (i) Keane’s envisaged min(f) = -0.835 for 50-dimensional problem is realizable; (ii) Liu-Lewis’ min(f) = -0.84421 for 200-dimensional problem is grossly sub-optimal.Computer programs (written by us in Fortran) are available on request.