Ant Colony Optimization with Stepwise Localization of the Discrete Search Space to Solve Function Optimization Problems

A new application of Ant Colony Optimization (ACO) called improved-EAS (i-EAS) is proposed for solving the function optimization problem. i-EAS is an improvement on the elitist ant system (EAS) which was devised to solve the Travelling Salesman Problem (TSP). It is capable of stepwise localization of the search space. Here, we examined methods that search for solutions in the real space Rn by approximating them using discrete values (binary data). Our proposed ACO uses stepwise localization of the search space to improve the accuracy of solutions. To localize the search space on the current step, our ACO searches for solutions recursively in neighbors of the best solution β found on the previous step. We assume that α is the number of times that the search space is localized, then we reduce the search space so that R(α) satisfies the equation R(α) = RANGE × (1 / 2)
^{(α × ln α)}
, where RANGE is provided as an initial value. Although the domain of each independent variable is reduced, the number of observable points is 2l and does not change according to α, which means that solution accuracy can be improved so that d(α) = RANGE × (1 / 2)
^{(α × ln α)+l-1}
, where d(α) is the interval of the observable data. To improve solution accuracy further, we perform mutation operations on the solution found by each ant every time it finishes a tour in search of solutions. Furthermore, in order to maintain population diversity, we dynamically altered the weight of elitist ant pheromone cyclically. The validity of i-EAS is verified by using well-known standard test functions with multiple peaks.