**4.3. Elitism, termination criteria and convergence**

The proposed algorithm also implements an elitism strategy, where the elite for **P**[k+1] is formed by selecting those individuals from both the elite of **P**[k] and the mutated elite of **P**[k] having the highest fitness value in the population. The mutation of the elite is performed with a probability *pm,e*=0.25*pm*. No crossover is performed on the elite.

This elitist model of the GA presents some convergence advantages over the standard GA. Using Markov chain modelling, it has been proved that GAs are guaranteed to asymptotically converge to the global optimum −with any choice of the initial population− if an elitist strategy is used, where at least the best chromosome at each generation is always maintained in the population (Bhandari, 1996).

The whole procedure is iterated until a termination criterion is satisfied. In our simulations, the search is terminated when there are no significant changes between the maximum and minimum values of the objective function in any two successive generations. Notice that, it can not be guaranteed that a valid solution is found in a finite number of iterations. Besides, the time required to compute an optimal solution increases exponentially with the size of the problem (Beckmann, 1999; Kunz, 1991; Funabiki, 1992). Thus, it is necessary to develop approximate methods capable of finding at least a near-optimum solution within a reasonable amount of time.

However, Bhandari et al. provided the proof that no finite stopping time can guarantee the optimal solution, though, in practice, the GA process must terminate after a finite number of iterations with a high probability that the process has achieved the global optimal solution. Note that, in our problem, the optimal string is not necessarily unique and there may be many strings that provide the optimal value (Bhandari, 1996).

In our proposed GA, the coding scheme guarantees that the traffic demand is always satisfied. However, in hard assignment instances it can be impossible to minimize the cost function to zero, i.e., some of the interference constraints may be violated by the generated assignment. In those cases where the optimal solution is not achieved in a finite time, invalid assignments should be resolved by manually assigning more frequencies to the affected cells (thus yielding a span that is larger that the lower limit). Nevertheless, none of the proposed test problem instances led to this situation.

At the end, the string *ui* corresponding to the highest fit chromosome is finally chosen as the channel allocation problem solution.
