Optimizing Rosenbrock Function Using Differential Evolution Algorithm

Differential Evolution (DE) is an efficient global optimization algorithm particularly suitable for solving nonlinear, multimodal optimization problems in continuous search spaces. This article demonstrates how to implement DE algorithm to optimize the classical Rosenbrock function, including key code implementation aspects such as population initialization and fitness evaluation.

The Rosenbrock function serves as a benchmark test function for optimization algorithms, featuring a narrow parabolic-shaped valley that causes many algorithms to converge to local optima rather than the global minimum. DE algorithm effectively explores such complex search spaces through its unique mutation, crossover, and selection operations. In code implementation, the mutation operation typically involves calculating difference vectors between randomly selected individuals, while crossover combines parent and mutant vectors to generate trial solutions.

The core algorithmic concept utilizes difference vectors between population individuals to generate new solutions, gradually approaching the global optimum through iterative updates. For Rosenbrock function optimization, the algorithm must handle strong coupling between variables - a strength of DE where mutation strategies like DE/rand/1 can effectively navigate interdependent parameters. Code implementation requires maintaining a population matrix where each row represents an individual solution vector.

During implementation, critical parameters require careful configuration: population size (typically 5-10 times problem dimensionality), mutation factor (F ∈ [0.4,1.0]), and crossover probability (CR ∈ [0.1,1.0]). Algorithm convergence speed and precision heavily depend on these settings. For Rosenbrock function, higher mutation factors (F ≈ 0.8-1.0) enhance global exploration capability. The selection operation compares trial vectors with target vectors using the Rosenbrock function as fitness evaluator: f(x) = Σ[100(x_i+1 - x_i²)² + (1 - x_i)²].

Validation results confirm that DE effectively locates Rosenbrock's global minimum (at x_i = 1, f(x) = 0), demonstrating strong performance in complex optimization scenarios. Compared to other algorithms, DE shows superior ability to avoid local optima through its differential mutation mechanism that maintains population diversity throughout evolution cycles.