Keywords: global optimization, hybrid optimization algorithms, particle swarm optimization, Powell method, heuristic algorithms, test functions.

In this paper, two hybrid auto-regulated optimization algorithms are developed, combining two proposed variations of the particle swarm optimization (PSO) [1] algorithm with a modified Powell algorithm [2]. The PSO algorithm is used to cover the entire search space, identifying the region of the global minimum, while the mathematical algorithm, starting from a point inside this region, quickly reaches the global minimum. This strategy increases reliability in comparison with mathematical methods, because it is more likely to reach the global minimum, and increases efficiency in comparison with pure heuristic algorithms.

One known drawback of heuristic algorithms is the necessity for user-defined, problem-dependent algorithmic parameters. In the present paper, two variations of the PSO algorithm are proposed, in order to make the algorithm performance less sensitive to user-defined parameters. In these variations, the particles are divided into four groups, each group using one set of algorithmic parameters. In one, more simple algorithm, these parameters are kept constant during all iterations. This increases the diversity of the swarm movement as a whole, since each group shows different behavior moving in a slightly different form. This algorithm is not truly auto-regulated, but it is shown to be more robust and less sensitive to user-defined parameters. The second is an auto-regulated algorithm, where the PSO parameters are up-dated following pre-defined rules, in an attempt to find an "optimal" set of parameters during problem solution. The two PSO variations developed use literature-suggested parameters, requiring no special parameter definition by the user. The two PSO variations proposed in this paper are shown to be less likely to remain trapped in regions of local minima, hence to be more reliable than the original or the constricted PSO algorithms.

The hybrid algorithms developed herein are employed in the solution of seven test problems from the literature. They are shown to be reliable and to out-perform nine out of ten algorithms from the literature, including the ant colony and simulated annealing algorithms, especially for test functions with several local minima. Comparing the two algorithms developed in this paper, the auto-regulated PSO algorithm is shown to perform better than the algorithm with constant parameters. In addition to the excellent results obtained with the algorithms developed, the general ideas presented in this paper can be used to develop several other hybrid algorithms, combining other heuristics with other mathematical programming algorithms.

1

D. Bratton, J. Kennedy, "Defining a standard for particle swarm optimization", Proceedings of the IEEE Swarm Intelligence Symposium, 120-127, 2007. doi:10.1109/SIS.2007.368035

2

M.J.D. Powell, "An efficient method for finding the minimum of a function of several variables without calculating derivatives", The Computer Journal, 7(2), 155-162, 1964. doi:10.1093/comjnl/7.2.155

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