Multimodal Optimization with the Local Optimum Ranking 2 Algorithm

Abstract

\n In optimization tasks, it is interesting to achieve a set of efficient solutions instead of one single output, in the case the best solution is not suitable. Many niching methods offer a diversified response, yet some important problems are common: (1) The most interesting solutions of each local optimum are not identified. Thus, the output is the overall population of solutions, which increases the work of the designer in verifying which solution is the most interesting. (2) Existing niching algorithms tend to distribute the solutions on the most promising regions, over-populating some local optima and sub-populating others, what leads to a poor optimization.To solve these challenges, a novel niching method is presented, named local optimum ranking 2. This sorting methodology favors the exploration of a defined number of local optima, and ranks each local population by objective value within each local optimum. Thus, is performed a multi-focus exploration, with an equalized number of solutions on each local optimum, while identifying which solutions are the local apices. Experimental results demonstrate the local optimum ranking 2 provides superior performance than other popular niching methods, for the selected test functions and global optimization algorithms. Also, its versatility is demonstrated in the several ways it can be combined with some of the most well-known methods.In a second experiment, the LOR2 algorithm is applied in the design optimization of a metallic cantilever beam. It is exemplified how the LOR2 algorithm can achieve a set of efficient and diverse design configurations, identifying which are the apices of each local optimum. Thus, the LOR2 facilitates multimodal optimization tasks, while offering both performance and diversity for design challenges.In addition, a third experiment describes how the algorithm can be applied to segment the domain of any function, with any type of input distribution or number of coordinates, into a mesh of similar sized or custom sized elements. Thus, it can segment a response surface named Kriging, significantly simplifying it and reducing computation time.