GNGTS 2018 - 37° Convegno Nazionale

GNGTS 2018 S essione 3.3 723 function value. Based on their power, some of the best initial countries become imperialists and start taking control of other countries (called colonies). The main operators of this algorithm are assimilation and revolution. Assimilation makes the colonies of each empire get closer to their imperialist. In addition to the assimilation, the position of some countries is randomly changed by using a parameter called revolution rate. This revolution process is used to preserve randomness of the algorithm. During assimilation and revolution, a colony might reach a better position and has the chance to take the control of the entire empire and replace the current imperialist state of the empire. Imperialist competition is another part of this algorithm, in which all the empires try to take possession of colonies of other empires. In each iteration, based on their power, all the empires have a chance to take control of one or more of the colonies of the weakest empire. These optimization tools are applied in each iteration until a stop criterion is satisfied. Particle swarmoptimization is an evolutionary computation technique developed by Eberhart and Kennedy (1995) that is inspired by social behaviour of bird flocking and fish schooling. In the classical version of PSO the set of candidate solutions to the optimization problem is defined as a swarm of particles which may flow through the parameter space defining trajectories which are driven by their own and neighbours’ best performances. PSO has undergone a plethora of changes since its development. One of the recent developments in PSO is the inclusion of Quantum laws of mechanics within the PSO framework: this leads to the QPSO (Sun et al. 2004). In PSO, a particle is stated by its position and velocity vectors, which determine the trajectory of the particle. However, if we consider quantum mechanics, the term trajectory is meaningless, because position and velocity of a particle cannot be determined simultaneously according to uncertainty principle. Indeed, each single particle in QPSO is treated as a spin-less one moving in quantum space and the probability of the particle’s appearing at position x i in the model space is determined from a probability density function. Simply speaking, the QPSO algorithm allows all particles to move under quantum-mechanical rules rather than the classical Newtonian random motion. The QPSO algorithm has simpler evolutional equation forms and fewer parameters than the standard PSO, substantially facilitating the control and convergence in the search space. Tests on analytic objective functions. We now compare the three algorithms on analytic test functions as the number of model parameters increases. We consider 2, 4, 6, 8 and 10 unknowns and for each dimension and for each method we perform 20 different runs from which we count the number of model evaluations requested to find the global minimum within a maximum number of evaluated model (10 6 ) and within a previously selected accuracy (Sajeva et al., 2017). The first test function we use is the Rastrigin function equals to: (1) where n is the dimension of the model space. The global minimum is located at [0, …, 0] n and is surrounded by regularly distributed local minima. For this function the initial ensemble of candidate solutions in each method is 10 times the model space dimension, whereas the accuracy is set at 0.1. In Fig. 1a we observe that all the methods successfully converge for all the dimensions, although as expected the number of model evaluations requested increases as the dimension of the model space increases. FA and QPSO show very similar performances, whereas ICArequires a much higher number of model evaluations to attain convergence. In other words, ICA seems characterized by a slower convergence rate than the other two algorithms. The second test function is the Schwefel function: (2) This is an extremely difficult function to optimize, because the local minima are irregularly