TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generated

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Second-order correction reduces the active constraint residual to cubic order

Statement

Let $c_{A} : U \to R^{r}$ with $r \geq 1$ be $C^{2}$ on a convex ball about $x$ , and assume $∥ D^{2} c_{A} (z) [u, v] ∥_{2} \leq M ∥ u ∥_{2} ∥ v ∥_{2}$ throughout that ball. Let $A = D c_{A} (x)$ have smallest singular value at least $s_{0} > 0$ , and let $p$ be small enough that $x + p$ and the correction segment stay in the ball. If $c_{A} (x) + A p = 0$ and $w$ is the correction of Minimum-norm second-order correction for identified active constraints, then $∥ w ∥_{2} \leq \frac{M}{2 s_{0}} ∥ p ∥_{2}^{2}, ∥ c_{A} (x + p + w) ∥_{2} = O (∥ p ∥_{2}^{3}) (p \to 0) .$ The correction is the unique minimum-norm vector satisfying $A w = - c_{A} (x + p)$ .

Facts & Assumptions

Given: The $C^{2}$ active constraint map, bounded second derivative, uniform row rank, and a step satisfying the linearized active equations.

[F1]

Minimum-norm second-order correction for identified active constraints sets $w = - A^{T} (A A^{T})^{- 1} c_{A} (x + p)$ and proves it is the minimum-norm solution of $A w = - c_{A} (x + p)$ .

[F2]

Second-order Taylor expansion $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} h^{T} H_{f} (a) h + o (∥ h ∥^{2})$ gives the second-order Taylor expansion with a remainder $o (∥ p ∥_{2}^{2})$ for $C^{2}$ maps, componentwise.

[A1]

The smallest singular value of $A$ is at least $s_{0} > 0$ , the second derivative bound is $M$ , and the trial and correction segments lie in the convex ball.

Proof

1.1givenF1F2algebra

Taylor's formula along $x + t p$ and the bound on $D^{2} c_{A}$ give $c_{A} (x + p) = c_{A} (x) + A p + R_{p} = R_{p}, ∥ R_{p} ∥_{2} \leq \frac{M}{2} ∥ p ∥_{2}^{2} .$ Since $∥ A^{T} (A A^{T})^{- 1} ∥_{2} = 1 / s_{\min} (A) \leq 1 / s_{0}$ , [F1] yields $∥ w ∥_{2} \leq M ∥ p ∥_{2}^{2} / (2 s_{0})$ and $c_{A} (x + p) + A w = 0$ .

2.1givenF1step 1.1algebra∎

Put $y = x + p$ . Taylor expansion at $y$ gives $c_{A} (y + w) = c_{A} (y) + D c_{A} (y) w + R_{w} = (D c_{A} (y) - A) w + R_{w},$ where $∥ D c_{A} (y) - A ∥_{2} \leq M ∥ p ∥_{2}$ and $∥ R_{w} ∥_{2} \leq M ∥ w ∥_{2}^{2} / 2$ . Therefore $∥ c_{A} (y + w) ∥_{2} \leq M ∥ p ∥_{2} ∥ w ∥_{2} + \frac{M}{2} ∥ w ∥_{2}^{2} \leq \frac{M^{2}}{2 s_{0}} ∥ p ∥_{2}^{3} + \frac{M^{3}}{8 s_{0}^{2}} ∥ p ∥_{2}^{4},$ which is $O (∥ p ∥_{2}^{3})$ for small $p$ . The minimum-norm assertion is [F1].

Depends on

Used by

A full SQP step is rejected by the L1 merit test and repaired by SOC Example

Dependency tree · two levels

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Sources

Jorge Nocedal and Stephen J. Wright, Numerical Optimization, 2nd ed. (standard reference, not scraped)