Chaotic Dynamics on a Simplex and Global Optimization Methods with Normalized Constraints

The application of chaos to global optimization methods are, 1) maps with respect to inner variables derived by discretizing gradient method models with Euler’s method are unstabilized by setting their sampling time large, 2) chaotic trajectories of the optimizer’s variables confined in the bounded searching domain are generated by nonlinear transformations of the unstabilized inner variables, and 3) the chaotic annealing method is available which conversely stabilize their dynamics by gradually decreasing the sampling time of them. However, the efficiency of the above-mentioned method called the chaotic global optimization was only reported for optimization problems constrained by upper and lower bounds.

In this paper, to the contrary, we attempt to apply the method to optimization problems with normalized equality and non-negativity constraints. First, based on the replicator model which is regarded as the gradient projection method with a variable metric, two types of chaotic maps on a simplex are presented. The one is a discretized steepest gradient model with respect to inner state variables to which the replicator model is equivalently transformed, and the other is a discretized replicator model for an unconstrained problem with respect to inner state variables obtained by variable transformation. Secondly, the bifurcation with respect to the sampling time of Euler’s method is shown and the feasibility of the trajectories on a simplex for normalized equality and non-negativity constraints is certified. Lastly, the chaotic global optimization methods with the annealing procedure are demonstrated in numerical simulations for a few constrained optimization problems.