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Local Quadratic Convergence of Sequential Quadratic Programming (SQP)

Exploring the cinematic intuition of Local Quadratic Convergence of Sequential Quadratic Programming (SQP).

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The Formal Theorem

Let f:Rn→R f: \mathbb{R}^n \to \mathbb{R} and h:Rn→Rm h: \mathbb{R}^n \to \mathbb{R}^m be twice continuously differentiable. Let x∗ x^* be a local solution satisfying the Second-Order Sufficient Conditions (SOSC) and Linear Independence Constraint Qualification (LICQ) with associated Lagrange multipliers λ∗ \lambda^* . If the SQP iterates {xk,λk} \{x_k, \lambda_k\} start sufficiently close to {x∗,λ∗} \{x^*, \lambda^*\} , then the sequence converges quadratically to {x∗,λ∗} \{x^*, \lambda^*\} , satisfying:
∣∣(xk+1,λk+1)−(x∗,λ∗)∣∣≤C∣∣(xk,λk)−(x∗,λ∗)∣∣2 ||(x_{k+1}, \lambda_{k+1}) - (x^*, \lambda^*)|| \leq C ||(x_k, \lambda_k) - (x^*, \lambda^*)||^2

Analytical Intuition.

Picture a high-speed vehicle navigating a winding, constrained mountain pass. Initially, the car is far from the optimal line, but as it nears the apex, it engages a predictive engine—the SQP algorithm. SQP doesn't just guess; it constructs a local quadratic model of the Lagrangian function, effectively creating a 'mini-map' of the curvature near the current point. By solving this quadratic subproblem, the algorithm mimics Newton’s method for finding roots of the optimality conditions. Because the quadratic approximation captures the second-order information (the Hessian) of the objective and constraints, the error in the position xk x_k and multipliers λk \lambda_k doesn't just shrink—it gets squared in every iteration. This is the hallmark of 'quadratic convergence': if your current error is 10−2 10^{-2} , the next step will likely drop it to 10−4 10^{-4} , then 10−8 10^{-8} . It is the mathematical equivalent of gaining 'superpowers' once you are in the neighborhood of the solution, turning a complex nonlinear search into a rapid, precise descent toward the optimal coordinate.
CAUTION

Institutional Warning.

Students often conflate 'quadratic convergence' of the SQP algorithm with the 'quadratic subproblem' it solves. The subproblem is quadratic by design, but the convergence rate is quadratic because the derivative of the optimality conditions vanishes at the solution, mimicking Newton's method behavior in unconstrained space.

Academic Inquiries.

01

Why is the convergence only 'local'?

The quadratic model is only a valid approximation near the optimum. Far from the solution, the curvature might be misleading, requiring globalization strategies like line searches or trust regions to ensure stability.

02

What happens if LICQ is violated?

If the constraints are not linearly independent, the Lagrange multipliers may not be unique or well-defined, causing the Jacobian of the KKT system to become singular and destroying the quadratic convergence rate.

Standardized References.

  • Definitive Institutional SourceNocedal, J., & Wright, S. J., Numerical Optimization.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Local Quadratic Convergence of Sequential Quadratic Programming (SQP): Visual Proof & Intuition. Retrieved from https://nicefa.org/library/fundamentals-of-optimization/local-quadratic-convergence-of-sequential-quadratic-programming--sqp-

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