Evolutionary and Swarm Intelligence Algorithms by Jagdish Chand Bansal Pramod Kumar Singh & Nikhil R. Pal

Author:Jagdish Chand Bansal, Pramod Kumar Singh & Nikhil R. Pal
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

The resulting feasible objective space is shown in Fig. 8. It is clear from this figure that the unconstrained Pareto-optimal region (shown by dashes) is now infeasible. The periodic nature of the constraint boundary makes the Pareto-optimal region discontinuous, having a number of disconnected continuous regions. The task of an optimization algorithm would be to find as many such disconnected regions as possible. The number of such regions can be controlled by increasing the value of the parameter b. It is also clear that with the increase in number of disconnected regions, an algorithm will have difficulty in finding representative solutions in all disconnected regions.

The above problem can be made more difficult by using a small value of d, so that in each disconnected region there exists only one Pareto-optimal solution. Figure 9 shows the feasible objective space for and (while other parameters are the same as that in the previous test problem). Although most of the search space is feasible, near the Pareto-optimal region the feasible search regions are disconnected, with finally each sub-region leading to a singular feasible Pareto-optimal solution. An algorithm will face difficulty in finding all discrete Pareto-optimal solutions because of the changing nature from a continuous to a discontinuous feasible search space near the Pareto-optimal region.

Fig. 8The constrained test problem CTP2

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