MATLAB Programming Example of SQP Algorithm

SQP (Sequential Quadratic Programming) algorithm is one of the efficient methods for solving nonlinear constrained optimization problems. Its core concept involves constructing and solving a quadratic programming (QP) subproblem at each iteration to approximate the optimal solution of the original problem. MATLAB, with its powerful numerical computation capabilities, serves as an ideal tool for implementing the SQP algorithm.

### Basic Workflow of SQP Algorithm Initialization: Define starting point, convergence tolerance, and maximum iteration count. QP Subproblem Construction: At the current iteration point, build the quadratic programming subproblem using the gradient of the objective function, Hessian matrix, and Jacobian matrix of constraints. QP Subproblem Solution: Invoke QP algorithm subroutine to obtain search direction. Line Search: Determine step size to ensure objective function reduction and constraint feasibility. Iteration Point Update: Update current solution using search direction and step size. Convergence Check: Verify if gradients, constraint violations meet termination criteria; otherwise return to step 2.

### Key Role of QP Subroutine The QP subroutine serves as the core module of SQP algorithm for solving quadratic programming subproblems. Implementation typically uses interior-point methods or active-set methods. In MATLAB, you can utilize built-in function `quadprog`, or develop custom QP solvers based on Lagrange multipliers with proper Karush-Kuhn-Tucker (KKT) condition handling.

### MATLAB Implementation Considerations Function Gradient and Hessian Calculation: Recommended to use symbolic toolbox or automatic differentiation tools (like `gradient` or `fmincon` with gradient options) for improved accuracy, avoiding manual derivation errors. Constraint Handling: Need to transform inequality and equality constraints into standard QP subproblem formulation using slack variables or direct incorporation. Efficiency Optimization: For large-scale problems, implement sparse storage for Hessian matrices, or use quasi-Newton methods (like BFGS) for Hessian approximation through `optimoptions` settings in `fmincon`.

By properly designing the interaction logic between main program and QP subroutine, the SQP algorithm can efficiently handle complex nonlinear optimization problems in engineering applications, with MATLAB's `fmincon` function providing built-in SQP implementation options through algorithm selection.