On deciding when to stop metaheuristics : properties, rules and termination conditions

Most metaheuristics lack a termination condition based on reasonable premises and guaranteeing the quality of the solution provided by the algorithm. We propose a methodological frame that distinguishes the concepts of properties of the final incumbent solution, rules and termination conditions. The frame is applied to discuss the use of popular stopping rules (such as maximum number of iterations and maximum number of iterations without improvement) in the resolution of global optimization and combinatorial optimization problems via random restart metaheuristics. This allows finding simple formulas for determining the maximum number of iterations corresponding to diverse combinations of problem types and properties desired for the final incumbent solution. The suggestions for further research include the application of the probabilistic theory of records to the study of terminal conditions for metaheuristics.