Chaotic Optimization Algorithm for Function Extremum Calculation

The Chaotic Optimization Algorithm (COA) is a global optimization method based on chaos theory. By incorporating the ergodicity and randomness of chaotic motion into the optimization process, it effectively escapes local optima and locates the global optimal solution. In code implementation, this involves initializing chaotic variables and iteratively updating their positions using chaotic maps.

The algorithm's core approach consists of two phases: first, a coarse search phase where chaotic variables explore the entire search space, followed by a fine search phase concentrating around promising solutions found during coarse search. The Logistic map is the most commonly used method for generating chaotic sequences, which can be implemented using the simple nonlinear iterative formula: x_n+1 = μx_n(1-x_n), where μ is a control parameter (typically 4) and x_n ∈ (0,1). This produces pseudorandom sequences exhibiting sensitivity to initial conditions and ergodicity properties.

In function extremum calculation, the algorithm first maps chaotic variables to the target function's domain using linear transformation. Each point's function value is evaluated, and the best solution is recorded. As iterations progress, the search range gradually narrows around the optimal solution while chaotic perturbations prevent premature convergence. Code implementation typically involves maintaining a best-so-far solution and dynamically adjusting search boundaries based on iteration count.

Compared to traditional optimization algorithms, chaotic optimization doesn't require gradient calculations and has lower requirements for function continuity and differentiability, making it particularly suitable for multimodal function optimization. However, careful selection of chaotic sequences and parameter tuning (such as the Logistic map parameter μ and iteration control parameters) is necessary to balance exploration and exploitation. Implementation often includes parameter sensitivity analysis to optimize performance.