Obstacles and difficulties for robust benchmark problems: A novel penalty-based robust optimisation method

This paper first identifies a substantial gap in the literature of robust optimisation relating to the simplicity, low-dimensionality, lack of bias, lack of deceptiveness, and lack of multi-modality of test problems. Five obstacles and difficulties such as desired number of variables, bias, deceptiveness, multi-modality, and flatness are then proposed to design challenging robust test problems and resolve the deficiency. A standard test suit of eight robust benchmark problems is proposed along with controlling parameters that allow researchers to adjust and achieve the desired level of difficulty. After the theoretical analysis of each proposed test function, a robust particle swarm optimisation (RPSO) algorithm and a robust genetic algorithm (RGA) are employed to investigate their effectiveness experimentally. The paper also inspects the effects of the proposed controlling parameters on the difficulty of the test problems and the proposal of a novel penalty function to penalize the solutions proportional to their sensitivity to perturbations in parameters. The results demonstrate that the proposed test problems are able to benchmark the performance of robust algorithm effectively and provide different, controllable levels of difficulty. In addition, the comparative results reveal the superior performance and merits of the proposed penalty-based method in finding robust solutions. Note: The source codes of the proposed robust test functions are publicity available at www.alimirjalili.com/RO.html.