fmincon Function for Solving Constrained Minimization Problems

In optimization problems, we often encounter situations where we need to find the minimum value of an objective function under specific constraints. MATLAB's Optimization Toolbox provides the `fmincon` function, specifically designed for solving multi-dimensional constrained optimization problems.

### 1. Core Functionality of fmincon `fmincon` is primarily used to solve nonlinear constrained optimization problems where either the objective function or constraints contain nonlinear relationships. It can handle various constraint types including: - Equality constraints (e.g., g(x) = 0) - Inequality constraints (e.g., h(x) ≤ 0) - Variable bound constraints (e.g., lb ≤ x ≤ ub)

### 2. Key Input Parameters The main inputs for this function include: - Objective function: The function to be minimized, typically defined as a MATLAB function handle - Initial guess: Starting values for optimization variables, which affect algorithm convergence - Constraints: Including linear constraints (A, b, Aeq, beq) and nonlinear constraints (defined through functions) - Variable bounds: Specifying lower and upper limits for variables to prevent optimization results from exceeding reasonable ranges

### 3. Algorithm Selection `fmincon` offers multiple optimization algorithms that users can select based on problem characteristics: - Interior-point method: Suitable for large-scale optimization problems, efficiently handling constraints - SQP (Sequential Quadratic Programming): Ideal for small to medium-sized problems with high convergence precision - Trust-region-reflective algorithm: Appropriate for scenarios with smooth objective functions but complex constraints

### 4. Output Result Analysis After optimization completion, `fmincon` returns the optimal solution for optimization variables, objective function value, convergence status, and other information. Users can use these results to determine whether optimization was successful and further evaluate solution stability.

### 5. Application Scenarios `fmincon` is widely used in engineering optimization, economic modeling, machine learning parameter tuning, and other fields requiring constrained optimization. Mastering its usage enables efficient solving of complex nonlinear programming problems.

Through appropriate parameter settings and algorithm options, `fmincon` can provide stable and efficient numerical solutions for constrained optimization problems.