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Research Papers

Enhanced speciation in particle swarm optimization for multi-modal problems
A derivation of the number of minima of the Griewank function

Enhanced speciation in particle swarm optimization for multi-modal problems

Huidae Cho, Dongkyun Kim, Francisco Olivera, Seth D. Guikema, 2011. Enhanced Speciation in Particle Swarm Optimization for Multi-Modal Problems. European Journal of Operational Research 213 (1), 15–23. doi:10.1016/j.ejor.2011.02.026 pdf R script

Abstract

In this paper, we present a novel multi-modal optimization algorithm for finding multiple local optima in objective function surfaces. We build from Species-based particle swarm optimization (SPSO) by using deterministic sampling to generate new particles during the optimization process, by implementing proximity-based speciation coupled with speciation of isolated particles, and by including “turbulence regions” around already found solutions to prevent unnecessary function evaluations. Instead of using error threshold values, the new algorithm uses the particle’s experience, geometric mean, and “exclusion factor” to detect local optima and stop the algorithm. The performance of each extension is assessed with leave-it-out tests, and the results are discussed. We use the new algorithm called Isolated-Speciation-based particle swarm optimization (ISPSO) and a benchmark algorithm called Niche particle swarm optimization (NichePSO) to solve a six-dimensional rainfall characterization problem for 192 rain gages across the United States. We show why it is important to find multiple local optima for solving this real-world complex problem by discussing its high multi-modality. Solutions found by both algorithms are compared, and we conclude that ISPSO is more reliable than NichePSO at finding optima with a significantly lower objective function value.

Highlights

Keywords: Particle swarm optimization; Metaheuristics; Multi-modal optimization; Rainfall characterization

A derivation of the number of minima of the Griewank function

Huidae Cho, Francisco Olivera, Seth D. Guikema, 2008. A Derivation of the Number of Minima of the Griewank Function. Applied Mathematics and Computation 204 (2), 694–701. doi:10.1016/j.amc.2008.07.009 pdf

Abstract

The Griewank function is commonly used to test the ability of different solution procedures to find local optima. It is important to know the exact number of minima of the function to support its use as a test function. However, to the best of our knowledge, no attempts have been made to analytically derive the number of minima. Because of the complex nature of the function surface, a numerical method is developed to restrict domain spaces to hyperrectangles satisfying certain conditions. Within these domain spaces, an analytical method to count the number of minima is derived and proposed as a recursive functional form. The numbers of minima for two search spaces are provided as a reference.

Keywords: Griewank function; Local minima; Optimization; Multi-modal optimization


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