Keane's bump function is considered as a standard benchmark for nonlinear constrained optimization. It is highly multimodal (nonconvex) and its optimum is located at the nonlinear constrained boundary. The paper optimizes Keane's function of different dimensions by the Repulsive Particle Swarm (RPS) and the Differential Evolution (DE) methods of global optimization. The RPS is endowed with intensive local search ability and DE uses the most recent formulation of crossover scheme. It is found that these methods, the DE in particular, are the most effective optimizers.

A visual appreciation of Keane's two-dimensional (m = 2) bump function may be obtained from the graphical presentation in Figure 1 (Hacker et al., 2002). As the dimension (m) grows larger, the optimum value of the function becomes more and more difficult to obtain. Keane (1994b) observed that for m = 20 the value of min[f(x)] could be about -0.76 and for m = 50 it could be about -0.835 but did not know this to be the case.

Keane's bump function is considered as a standard benchmark for nonlinear constrained optimization. It is highly multimodal and its optimum is located at the nonlinear constrained boundary. Emmerich (2005, p. 116) noted that the true minimum of this function is unknown. Using their various hybridized genetic algorithms Hacker et al. (2002) obtained min[f(x)] = -0.365 for a two-dimensional and -0.6737 for a 10-dimensional Keane's problem. They also found that the genetic algorithms (without hybridization) perform worse than their hybridized genetic algorithms.

This paper intends to optimize Keane's function of different dimensions by the Repulsive Particle Swarm (RPS) and the Differential Evolution (DE) methods of global optimization. RPS is endowed with intensive local search ability. Similarly, our DE uses the most recent (available) formulation of crossover scheme suggested by Kenneth Price. The computer programs in FORmula TRANslation (FORTRAN) have yielded very good results for quite varied and difficult problems (Mishra, 2006a, 2006b and 2006c).

First min[f(x)] for the two-dimensional (m = 2) Keane's problem were obtained and DE min[(f)] = -0.364979746 constraints g₁(x) = 0.0 and g₂(x) = -11.9306415 for x₁ = 1.60086044 and x₂ = 0.468498055. Against this, RPS min[(f)] = -0.364979123; g₁(x) = -3.82208509E-007; g₂(x) = -11.9310703 for x₁ = 1.60025376 and x₂ = 0.468675907. These results are comparable with the optimum value obtained by Hacker et al. (2002). The DE results are marginally better than the RPS results.

Keane's Bump Function, Repulsive Particle Swarm, RPS, Nonconvex Optimization, Differential Evolution methods, DE, FORmula TRANslation, FORTRAN, Genetic algorithms, Global optimization, Multidisciplinary Analysis, Adaptive Memetic Algorithms.