MATLAB Implementation of Simulated Annealing Algorithm with Code Examples

Simulated annealing is a heuristic optimization algorithm inspired by the annealing process in metallurgy. When implemented in MATLAB, this algorithm effectively solves complex optimization problems, particularly when the objective function contains multiple local optima.

### Algorithm Workflow Overview Initial Solution Generation: The algorithm starts with a random solution, which could be a randomly selected parameter vector or initial point. In MATLAB implementation, this is typically achieved using functions like rand() or randn() to generate random initial parameters. Temperature Setting: Initial temperature (high temperature) and cooling strategy (linear, exponential, or adaptive cooling) must be defined. The temperature controls the probability of accepting worse solutions during the search process. Code implementation often uses a temperature variable that decreases according to a predefined schedule. Neighborhood Search: Generate a new solution in the vicinity of the current solution. This step is implemented by randomly perturbing certain parameters of the current solution using small random variations or Gaussian noise. Acceptance Criterion: The Metropolis criterion determines whether to accept the new solution. If the new solution is better than the current one, it's accepted directly; otherwise, worse solutions are accepted with a certain probability to avoid local optima. This is typically coded using conditional statements and probability calculations based on exponential functions. Cooling Process: As iterations progress, temperature gradually decreases, reducing the probability of accepting worse solutions until the algorithm converges. The cooling schedule is implemented through a multiplicative factor applied at each iteration.

### Key Implementation Points in MATLAB Objective Function: Define an objective function (for minimization problems) that accepts a solution vector and returns a scalar value. This function is typically implemented as a separate MATLAB function file or anonymous function. Cooling Strategy: Commonly uses exponential cooling, where temperature is multiplied by a decay factor (0.9-0.99) at each iteration. This can be implemented using a simple multiplication operation within the main loop. Termination Conditions: Algorithm stopping can be based on temperature thresholds, maximum iterations, or solution quality improvement. These conditions are checked using while loops or conditional break statements in the MATLAB code.

### Applicable Problems and Optimization The simulated annealing algorithm suits both continuous and discrete optimization problems, especially effective for multimodal function optimization. In MATLAB implementation, balancing exploration (global search) and exploitation (local optimization) capabilities can be achieved by adjusting initial temperature, cooling rate, and neighborhood search range through parameter tuning.

For beginners, starting with simple two-dimensional optimization problems is recommended to observe how the algorithm escapes local optima and gradually approaches the global optimum solution. This helps in understanding the algorithm's behavior through visualization of the search process.