# Using Feedback Strategies in Simulated Annealing with Crystallization Heuristic and Applications
## 1 Introduction  
The solution of engineering problems often involves optimization. There are several well-known optimization methods, mainly based on the cost function gradient property. These methods require a seed to start the search and perform the search walking in the direction of the largest gradient [1]. However, in many applications, these methods cannot be used. The difficulties are that either the gradient property is not available, the gradient property is noisy, or the cost function is discrete. These methods are called deterministic methods because, starting from a given seed, the final result is always the same. The determination of the seed is also an additional task which eventually cannot be easily determined. To overcome problems such as the gradient property of the cost function and seed determination, a large set of metaheuristic methods have been proposed. Usually, metaheuristics mimic some emergent behavior from nature. Genetic Algorithms (GA) [2] model the dominance of stronger creatures over weaker creatures. GA has two conventionaloperators: crossover and mutation. Particle Swarm Optimization (PSO) [3] models the learning task, where creatures can learn from themselves or from others. Simulated annealing (SA) [4] is a stochastic metaheuristic which models metal annealing. Differential Evolution [5] tries to improve upon a current solution given a measure of quality. Several other metaheuristics have been proposed, each of them having a specific motivation, usually from nature. The metaheuristics do not need a seed, and they do not use any gradient property of cost functions. They are connected with the types of parameters and the cost function. For example, GA originally manipulated integer or limited precision parameters. Modifications in the original GA, such as increasing the number of operators, allow its usage even in combinatorial problems [6]. The PSO was originally proposed to deal with real parameters, and modifications in the original approach also allow its application to combinatorial problems [7]. Combinatorial problems are considered the most difficult type of problem--they are called NP-hard problems. SA was originally proposed to deal with combinatorial problems, such as traveling salesman and graph partitioning problems. It is the only metaheuristic that has a proof of convergence to the global optimum [8]. SA is the simplest algorithm with only two loops. However, it has been difficult to include real parameters in the SA. This paper explains the motivations towards the enhancement of SA to manipulate real parameters and the proposed SA with the crystallization heuristic. The crystallization heuristic does not interfere with combinatorial or integer parameters. SA with the crystallization heuristic uses feedback to control the parameter named the crystallization factor, which represents the parameter sensibility [9]. SA with the crystallization heuristic was successfully applied to problems with discrete cost functions [10; 11; 12]. This research improves the crystallization heuristic by combining the Bates and Gaussian distributions. Additionally, feedback strategies are proposed to emphasize the desired behavior through exploration or refinement. This proposal improves upon the results of the benchmark test from the literature. This paper starts by explaining the proposed SA with the crystallization heuristic. Section 2 also describes several proposals for the use of real parameters with SA and the motivation to propose SA with the crystallization heuristic. Section 3 presents the results of solving some benchmark problems with SA with the crystallization heuristic. Section 4 describes an extension of SA with the crystallization heuristic in which the cost function is partially evaluated. This is called interval SA with the crystallization heuristic. The study case solves the electrical impedance tomography problem. Section 5 explains that the crystallization factor is related to the parameter sensibility. This is an important property which can enhance the SA convergence.