MATLAB Implementation of Chaotic Optimization Algorithms

Chaotic optimization is a global optimization method based on chaotic theory that leverages the randomness and ergodicity of chaotic maps to optimize objective functions. MATLAB provides powerful numerical computation capabilities that are well-suited for implementing various chaotic optimization algorithms. The following sections describe several common chaotic optimization implementation approaches with code-related details: 1. Logistic Map Chaotic Optimization The Logistic map is one of the simplest chaotic systems with the iterative formula: x_{n+1} = mu * x_n * (1 - x_n), where mu controls chaotic behavior. In optimization, chaotic sequences generated by Logistic mapping can perturb the search space to avoid local optima. MATLAB implementation typically involves initializing parameters (mu ≈ 3.9-4.0) and iterating the map to generate chaotic variables for solution exploration. 2. Chen Chaotic System Optimization Chen chaotic system is a high-dimensional chaotic map with complex dynamic characteristics. In MATLAB, chaotic sequences can be generated by solving Chen system's differential equations using ODE solvers like ode45, which can then be applied for parameter adjustment in optimization algorithms. The implementation requires defining the three-dimensional differential equations and integrating them numerically. 3. Duffing Oscillator Chaotic Optimization Duffing oscillator is a nonlinear dynamic system that can generate chaotic signals. In optimization problems, Duffing oscillator's chaotic characteristics enhance search diversity, making it suitable for high-dimensional optimization problems. MATLAB implementation involves simulating the forced Duffing equation with appropriate parameters to produce chaotic behavior for optimization perturbations. 4. Henon Map Chaotic Optimization Henon map is a two-dimensional discrete chaotic system with iterative formulas involving nonlinear coupling of two variables. During optimization, Henon mapping improves search ergodicity, particularly for multimodal optimization problems. Code implementation requires simultaneous iteration of both variables (x and y) according to the Henon equations to generate two-dimensional chaotic sequences. 5. Lorenz System Chaotic Optimization Lorenz system is a classical three-dimensional chaotic system often used to test chaotic optimization algorithm performance. In MATLAB, ODE solvers can simulate Lorenz system's chaotic characteristics, which can be combined with optimization algorithms for parameter optimization. Implementation involves setting up the Lorenz equations with standard parameters (σ=10, ρ=28, β=8/3) and using numerical integration. Implementation Strategy Summary Chaotic Sequence Initialization: Select appropriate chaotic mapping to generate initial solutions through iterative computations. Chaotic Perturbation Search: Introduce chaotic variables during optimization process to enhance global search capability using random-like yet deterministic sequences. Adaptive Adjustment: Dynamically adjust chaotic parameters based on optimization progress to balance exploration and exploitation phases. Convergence Condition Optimization: Combine with traditional optimization methods (like PSO, GA) to improve convergence speed while maintaining global search properties. Chaotic optimization implementation in MATLAB typically combines numerical computation toolkits, such as using ode45 to solve chaotic differential equations or directly applying iterative formulas to generate chaotic sequences. The method's advantage lies in its ability to escape local optima, making it suitable for complex optimization problems with multiple local minima.