# Combining Parallel Stochastic Methods and Mixed Termination Rules in Optimization
## 2 Adopted Algorithms
### The Method Multistart
In each dimension, 10 different versions of the functions were used and average execution times and function calls were recorded. Also, the proposed method was applied on the GKLS function using

Figure 6: Wilcoxon rank-sum test results for the comparison of different termination rules as used in the proposed parallel optimization technique and for five processing units. different numbers of processing units. The average function calls from this experiment are show graphically in Figure 8 and the average execution times in Figure 9.

The graphs above show once again that adding processing units to the proposed technique can reduce not only the execution time, as expected, but also the required number of function calls. In the specific scheme, the execution time increases slightly for a certain number of execution units and above, as this time also includes the time spent by the execution engine for the synchronization of the processing units as well as for the exchange of information between them. Furthermore, in Figure 10, the behavior of speedup and efficiency is depicted as a function of the number of PUs. The execution of the algorithm with a single processing unit, where the speedup is 1, serves as the reference point. As the number of PUs increases, a corresponding increase in speedup is observed, though it is often less than the ideal (equal to the number of PUs) due to communication delays and overhead. Conversely, efficiency

Figure 8: Average function calls for the proposed method and different dimensions of the GKLS function. Figure 7: Experiments using the proposed parallel technique and the CM function. The dimension of the function \(n\) varies from 1 to 20 and the number of processing units changes from 1 to 20.

decreases with the increase in PUs, a common phenomenon in parallel computing, caused by non-linear scaling and uneven workload distribution. ## 5 Conclusions

In this paper, the use of mixed termination rules for global optimization methods was thoroughly presented, and a new global optimization method that takes full advantage of parallel computing structures was afterward proposed. The use of mixed termination rules leads a series of computational global optimization techniques to find the global minimum of the objective function faster, as was also shown by the experimental results. The termination rules exploited are based on asymptotic criteria and are general enough to be applicable to any global optimization problem and stochastic global optimization technique. Furthermore, the new stopping rule is utilized in a novel stochastic global optimization technique that involves some basic global optimization techniques. This method is designed to be executed in parallel computation environments. At each step of the method, a different global optimization method is also executed on each parallel computing unit, and these different techniques exchange optimal solutions with each other. In current work, the

Figure 10: Speedup and efficiency of the parallel approach. Figure 9: Average execution time for the GKLS function and the proposed method. global optimization techniques of Differential Evolution, Multistart and PSO were used in the processing units, but the method can be generalized to use other stochastic optimization techniques, such as genetic algorithms or simulated annealing. Also, the overall method terminates with the combined termination rule proposed here, and the experimental results performed on a series of well-known test problems from the relevant literature seem to be very promising. However, the present work is limited to a specific set of optimization methods and termination criteria. Therefore, future extensions of this work may include the integration of additional and more advanced termination criteria, as well as the implementation of alternative global optimization methods in the processing units, such as genetic algorithms or variations of simulated annealing.