# Simulated Annealing with Exploratory Sensing for Global Optimization
## 3 Simulated Annealing with Exploratory Sensing (SAES) Algorithm  
The objective of the proposed work is to increase the probability of finding a global optimum by using a memory-based mechanism. The GM works as a sensing search memory where the visited partitions are recorded to ensure the exploration of the search space. The search walker is then enforced to attend non-visited partitions after several iterations (Markov chains) while keeping a record of several best solutions found, as shown in the flowchart in Figure 2. Also, the search is carried out in two phases:  
1. **The exploration phase. ** This stage has several iterations within several Markov chains and aims to explore the constrained search space within the lower and upper limits of the solutions' values without affecting the way of how SA works in each Markov chain. Instead of starting the new Markov chain from the last accepted state, applying the GM directs the search to visit solutions in non-visited partitions of the search space. The cooling schedules and the acceptance criteria are not affected by applying the GM, while the temperature is changed in each Markov chain ones. Therefore, in each Markov chain, an intensification process is carried out in the new partition of the search space, while the search is keeping records of the best-found solutions. The adjustment of the SA settings should explore the search space in the first phase. Extensive exploration is maintained in our proposed method by a diversification index. The diversification index is a ratio of the number of visited partitions to the number of all partitions in the GM, and it is measured after each Markov chain. For instance, this phase can be ended when the diversification index reaches a pre-defined ratio \(\gamma\) of the partitions have been visited. The numerical experiments shown later indicate that the appropriate value for the parameter \(\gamma\) is 0.9, and this means visiting a percentage greater than 90% of the GM partitions. Another  
Figure 1: An example of the _Gene Matrix_ in \(R^{2}\). parameter called a diversification threshold \(\delta\) is applied to control the direction to un-visited partitions of the search space and this is called diversification sensing. This parameter is a smaller anticipated value of the diversification index. For example, the diversification threshold value \(\delta=0.04\), means that when the search in one Markov chain has not changed the diversification value by this threshold, the diversification process will be called to move to new search partitions. Otherwise, the diversification process is not used as the search is achieving good diversification using the escaping mechanism. 2. **The intensification phase. ** After reaching a specified level of diversification where most of the searching partitions were visited, the search is directed to start from the best-found state to do a more local search in that region. Although the diversification is supposed to be achieved mainly in the first phase using the GM, it is essential to ensure that the initial temperature is wisely chosen to avoid getting trapped in local minima earlier at the start of this phase. We choose to use a static cooling schedule that allows sufficient temperature after ending the exploration phase. The search is ended when reaching the maximum iteration, which equals the number of Markov chains multiplied by each Markov chain's length. Figure 2: Flowchart of the proposed methodology. After the two leading search phases are accomplished, a faster local search is called to refine the best solution obtained so far. This could improve the SA algorithm output since this algorithm is known to be reel around optimal solutions in the final stages of the search [67; 68]. It is worth noting that the two phases are executed sequentially and not overlapping together to maintain the main structure of the SA algorithm, especially the cooling scheduling. This is also so that the proposed algorithm does not lose the convergence behavior recognized in the standard SA algorithm.