# Using Feedback Strategies in Simulated Annealing with Crystallization Heuristic and Applications
## 3 Benchmark Results  
SA with the crystallization heuristic was applied to some benchmark functions. Table 2 shows the function definitions, with \(n\) being the number of variables. Table 3 shows all the function limits and global optima. Each function has a different feature that is evaluated in this benchmark. The sphere function is the easiest function, with a global optimum that is well defined. The Rosenbrock function has a great deal of difficulty finding the global optimum once it is located in a flat-shaped valley. The Rastrigin, Ackley, Griewangk, and Weierstrass functions have several local minima, in which the capacity of the algorithm to escape these points is tested. The Zakharov function has a global optimum in place with a low gradient, making it difficult to determine with algorithms that use the gradient. Figure 3 shows all the functions plotted in a two-dimensional form; i.e., all functions have two variables. All these functions are minimized with the proposed SA with the crystallization heuristic using two different positive feedback strategies: I and IV. The positive feedback strategy I resets the crystallization factor (\(c_{k}\gets 1\)) every time a candidate is accepted (see Table 1). The positive feedback strategy IV resets the crystallization during the exploration phase and decreases the crystallization factor by three in the refinement phase every time a candidate is accepted. The verification of whether the algorithm is at an exploration or refinement phase is done by evaluating the standard deviation of the costs. Each function is tested with three different numbers of variables: 10, 30, and 50. The maximum number of iterations is 10,000 times the number of variables. Each configuration is repeated 100 times, and the average and standard deviation of the cost are shown in Table 4. Strategy I is the positive feedback which resets the crystallization factor, and Strategy IV is the positive feedback strategy with a different strategy for the exploratory and refinement phase of the algorithm. The results show that the higher the number of variables, the worse the SA performs for both feedback strategies. Strategy IV performs better than Strategy I for all functions. This was expected, as every time the crystallization factor is restarted, it is necessary for several iterations to increase it to the point a new candidate is accepted. Then, the crystallization factor is kept low, making it difficult to refine to a better solution. The optimization performed by strategy I can achieve the results of strategy IV; however, a higher number of iterations will be necessary, making this strategy slower. The proposed SA approach performed better in the Rastrigin and Weierstrass functions, finding the exact global minimum. The Sphere and Ackley functions had an average performance, followed by the Griewangk function. The Rosenbrock and Zakharov functions had the worst performance, especially in the tests with 50 variables (the source code used here to evaluate the proposed method is available at [[https://github.com/LabGeocomp/sacryst](https://github.com/LabGeocomp/sacryst)]([https://github.com/LabGeocomp/sacryst](https://github.com/LabGeocomp/sacryst)), 1 November 2021).