Sqp

Sequential Quadratic Programming (SQP) is a class of iterative methods for nonlinear constrained optimization. Given a problem to minimize a scalar objective f(x) subject to equality constraints h(x)=0 and inequality constraints g(x)≤0, SQP solves a sequence of quadratic programming (QP) subproblems that approximate the original problem around the current iterate xk. The quadratic model uses the gradient of f and a Hessian (or approximation) of the Lagrangian, while the constraints are linearized at xk.

At iteration k, with Bk an approximation to the Hessian of the Lagrangian L(x, λ, μ) = f(x) + λ^T h(x) + μ^T g(x), the subproblem is typically written as: minimize 1/2 p^T Bk p + ∇f(xk)^T p subject to ∇h(xk) p = -h(xk) and ∇g(xk) p ≤ -g(xk). Here p is the step to be taken in x. After solving the subproblem, the next iterate is xk+1 = xk + p, and dual information (multipliers) is updated. To promote global convergence, a merit function or a line search or trust-region strategy is often employed to ensure sufficient decrease in the objective and progress toward feasibility.

Convergence properties: under standard regularity conditions and a positive-definite Bk on the constraint subspace, SQP exhibits fast local convergence, frequently superlinear or quadratic, and handles multiple nonlinear constraints effectively. Practical performance depends on the quality of the Hessian update and the efficiency of solving the QP subproblems.

Variants and related methods: Sequential Least Squares Programming (SLSQP) is a widely used variant that integrates least-squares formulations; other approaches combine SQP ideas with augmented Lagrangian or interior-point techniques. Applications span engineering design, control, trajectory optimization, economics, and more.