Sequential Quadratic Programming Algorithm

Sequential Quadratic Programming (SQP) is an efficient numerical optimization method for solving nonlinear constrained optimization problems. The algorithm's core principle involves iteratively approximating the original problem as a series of Quadratic Programming (QP) subproblems, progressively converging toward the optimal solution.

In MATLAB implementation, the SQP algorithm typically follows these key computational steps: First, initialize variables and parameters including initial guesses for design variables, constraint tolerances, and convergence criteria. The main iteration loop then begins, where at each iteration the Lagrangian function is constructed at the current point, with its Hessian matrix calculated or approximated using quasi-Newton methods like BFGS updating. The original problem is transformed into a QP subproblem featuring a quadratic objective function approximation and linearly approximated constraints.

Each QP subproblem solution provides a search direction, combined with line search or trust region strategies to determine step sizes that ensure objective function reduction while satisfying constraints. Convergence is verified by checking Karush-Kuhn-Tucker (KKT) conditions or monitoring changes in iteration step sizes. MATLAB implementations require careful attention to numerical stability through constraint violation regularization and Hessian matrix positive-definiteness maintenance.

For engineering applications, SQP effectively handles optimization problems with nonlinear equality and inequality constraints in fields like mechanical design and trajectory planning. MATLAB implementations typically utilize built-in Optimization Toolbox functions or custom coding combined with QP solvers like quadprog for subproblem resolution. Key implementation considerations include proper handling of constraint Jacobians, efficient Hessian updates, and robust convergence checks to ensure algorithm reliability across various problem types.