MOEO-EED: A multi-objective equilibrium optimizer with exploration–exploitation​ dominance strategy

The work proposes multi-objective variants of the recently-proposed equilibrium optimizer (EO) using an archive to obtain Pareto optimal solutions and a crowding distance approach to preserve the diversity among the non-dominated solutions. However, due to the use of constant values for the controlling parameters of EO, the exploration and exploitation of multi-objective EO (MOEO) are not accelerated, so the first variant is proposed with a number of linear and non-linear equations to generate increasing and decreasing values proportional to the number of iterations that will eventually improve exploratory and exploitative behaviors of MOEO. In the second proposed variant of MOEO with exploration–exploitation​ dominance strategy (MOEO-EED), solutions are updated according to the number of dominated solutions. If a solution has a high number of dominated solutions, it will go through fewer abrupt changes, while others will undergo major changes. In addition, a novel strategy known as a Gaussian-based mutation (G) strategy is proposed to use the Gaussian distribution to generate two different step sizes: small step sizes that are generated under a small sigma value for the Gaussian distribution to promote the exploitation capability, and high step sizes under a high sigma value to increase the exploration operator. The tradeoff between those two-step sizes is predefined through experimentation. This strategy is integrated to generate the third variant, namely MOEO-EED-G. In the fourth variant (OMOEO-EED-G), the tradeoff between the best solution selected from the archive and its opposite is achieved with a probability to increase the diversity and accelerate the convergence of MOEO-EED-G. Finally, the efficacy of the proposed algorithms is tested on four benchmark multi-objective functions to show that the proposed algorithms, especially OMOEO-EED-G, are superior to selected state-of-the-art multi-objective algorithms.