Genetic Algorithm Implementation in MATLAB for Univariate, Multivariate Unimodal, and Multivariate Multimodal Functions

In scientific computing and engineering optimization, Genetic Algorithm (GA) is a global optimization method based on natural selection and genetic mechanisms. MATLAB provides a flexible Genetic Algorithm toolbox that effectively solves optimization problems for univariate functions, multivariate unimodal functions, and multivariate multimodal functions. Univariate Function Optimization Univariate function optimization represents one of the simplest applications of genetic algorithms, ideal for verifying algorithm convergence. By implementing appropriate encoding schemes (such as binary or real-value encoding) and fitness functions, the genetic algorithm can locate global optima within the defined domain. For example, when solving extremum points of univariate quadratic functions, the algorithm quickly identifies vertex positions even when local extremum points exist. In MATLAB implementation, the ga function can be configured with lower and upper bounds while defining the fitness function using function handles. Multivariate Unimodal Function Optimization For multivariate unimodal functions (such as the Rosenbrock function), genetic algorithms gradually approach the optimal solution through population iterations. The key lies in adjusting crossover probability, mutation probability, and population size to ensure efficient exploration of the search space. MATLAB's Global Optimization Toolbox offers default parameter settings while supporting user customization to improve convergence speed. Implementation typically involves specifying the number of variables and using vectorized fitness function evaluations for better performance. Multivariate Multimodal Function Optimization Multivariate multimodal functions (like the Rastrigin function) contain multiple local extremum points, demanding stronger global search capabilities from genetic algorithms. The algorithm must maintain population diversity to avoid premature convergence to suboptimal solutions. Adaptive mutation strategies or hybrid approaches combining other optimization methods (such as simulated annealing) can enhance the ability to escape local optima. MATLAB implementations may utilize custom mutation functions and hybrid function options to improve global search performance. Important Considerations Regardless of the function type being optimized, parameter tuning and termination condition settings are crucial. Appropriate maximum iterations and fitness tolerance values balance computational efficiency with precision. Additionally, visualizing the iteration process (such as fitness curves) facilitates algorithm performance analysis. MATLAB's plotting capabilities provide convenient support for this purpose. The gaoptimset function allows detailed configuration of algorithm parameters including display options for monitoring convergence. Through practical case studies, one can gain more intuitive understanding of genetic algorithm performance differences across various function types and their applicable scenarios. MATLAB's built-in examples and demonstration scripts offer excellent starting points for implementing these optimization techniques.