MATLAB Code Implementation for Finding Function Minimum Values

Solving unconstrained nonlinear function minimization is a common numerical optimization problem in MATLAB. MATLAB provides multiple built-in functions to achieve this goal, suitable for functions with different characteristics and complexities. Here are the main implementation approaches and methods: `fminunc` Function The `fminunc` function is the core optimization tool in MATLAB's Optimization Toolbox for unconstrained minimization of smooth nonlinear functions. Users need to provide the objective function and an initial guess, and the function automatically selects appropriate algorithms (such as quasi-Newton methods or trust-region methods) for iterative solving. If the function is differentiable, gradient information can be provided to accelerate convergence. Implementation typically involves defining the objective function as a separate function file or anonymous function. `fminsearch` Function Based on the Nelder-Mead simplex method, this function is suitable for non-differentiable functions or functions with noise. This derivative-free method compares function values to gradually approach the minimum point, though it has slower convergence and is best for low-dimensional problems. The code implementation requires only the objective function and initial point without gradient calculations. Global Optimization Solutions For functions with multiple local minima, you can use `GlobalSearch` or `patternsearch` functions from the Global Optimization Toolbox. These employ multi-start search or pattern search techniques to increase the probability of finding global minima. Implementation involves creating an optimization problem object and configuring solver settings. Parameter Tuning and Output Parameters like tolerance levels and maximum iterations can be set using `optimoptions`. The output includes the minimum point, function value, and convergence status, helping users verify result reliability. For example: options = optimoptions('fminunc','Display','iter','MaxIterations',1000); For complex problems, it's recommended to first visualize the function graph to determine initial points, then select appropriate methods based on function characteristics. MATLAB's documentation provides additional extended functionalities such as Hessian matrix calculations or parallel computing support.