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 (RPS) and the Differential Evolution (DE) methods of global optimization. The DE optimization program has gone a long way to obtain the optimum (if so!) results. Application of the RPS program has clearly failed to minimize the Keane's function of any considerable size. Then we have conjectured that the values of the decision variables (of Keane function) 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. Incorporating our conjecture into computation, in each iteration we arrange the variable values in a descending order and then evaluate the function. This modification has improved the RPS performance greatly. The performance of DE also is significantly enhanced. 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 by the RPS and the DE; (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