Sequential Quadratic Programming

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

Sequential Quadratic Programming (SQP) is one of the most effective methods for solving nonlinear programming (NLP) problems, possessing both global convergence and local superlinear convergence properties. Its fundamental idea is to convert the original NLP problem into a sequence of quadratic programming (QP) subproblems.

Core Elements

Problem Formulation

A general NLP problem can be stated as:

\min \quad J = F(y)

\text{s.t.} \quad h_i(y) = 0, \quad i = 1, 2, \ldots, r

g_j(y) \leq 0, \quad j = r+1, r+2, \ldots, s

where $y$ includes both state variables and control variables.

Lagrangian Function

The Lagrangian of the NLP problem is:

L(y, \lambda) = F(y) + \sum_{i=1}^{r} \lambda_i h_i(y) + \sum_{j=r+1}^{s} \lambda_j g_j(y)

QP Subproblem

At each iteration, the constraints are approximated by a first-order Taylor series and the objective function by a second-order Taylor series, converting the NLP problem into a QP subproblem:

\min_{d_k} \nabla F(y_k)^{\mathrm{T}} d_k + \frac{1}{2} d_k^{\mathrm{T}} H_k d_k

where $H_k$ is an approximate positive-definite form of the Lagrangian Hessian matrix, updated using the modified BFGS formula:

H_{k+1} = H_k + \frac{s_k s_k^{\mathrm{T}}}{s_k^{\mathrm{T}} \delta_k} - \frac{(H_k \delta_k)(H_k \delta_k)^{\mathrm{T}}}{(H_k \delta_k)^{\mathrm{T}} \delta_k}

Iteration Steps

Given an initial point $y_0$ , initial matrix $H_0$ , and tolerance $\varepsilon$
Solve the QP subproblem to determine the search direction $d_k$ and Lagrange multipliers $\lambda_k$
Determine the step size $\alpha_k$ via exact line search, and compute the new iterate $y_{k+1} = y_k + \alpha_k d_k$
If $\|y_{k+1} - y_k\| \leq \varepsilon$ , stop; otherwise update $H_k$ and return to step 2

Comparison with Other Optimization Methods

Method	Characteristics	Applicable Scenarios
SQP	Gradient-based, fast convergence, local optimum	Continuously differentiable NLP problems
Genetic Algorithm	Global search, gradient-free	Discrete or non-smooth problems
Simulated Annealing	Global search, avoids local optima	Complex multimodal problems

Application Value

SQP is one of the core methods for trajectory optimization. In launch vehicle and ballistic missile trajectory optimization, SQP is used to solve NLP problems where the objective function is payload capacity or impact accuracy, subject to orbital insertion conditions and path constraints. Its fast convergence makes it widely used in engineering design.

Newton's Iteration Method

References

Zheng Wei, An Xueying, Zhou Xiang, He Ruizhi. Aerospace Flight Mechanics (空天飞行力学)[M]. National University of Defense Technology, 2026.
Jia Peiran, Chen Kejun, et al. Long-Range Rocket Ballistics (远程火箭弹道学)[M]. National University of Defense Technology Press.