ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A quadratic equality model shows the AL conditioning tradeoff

Example

For $f (x, y) = \frac{1}{2} (x^{2} + y^{2})$ subject to $h (x, y) = y = 0$ , the equality augmented Lagrangian with penalty $ρ > 0$ and multiplier $ν$ has Hessian $diag (1, 1 + ρ)$ . Its exact inner minimizer is $(0, - ν / (1 + ρ))$ , and the multiplier update sends $ν$ to $ν / (1 + ρ)$ . Thus increasing $ρ$ improves the exact multiplier contraction factor while worsening the inner Hessian condition number to $1 + ρ$ .

Facts & Assumptions

Given: The equality augmented-Lagrangian formula and multiplier update, with fixed $ρ > 0$ .

[F1]

Equality and inequality augmented Lagrangians and multiplier updates defines $L_{ρ}^{E} = f + ν h + \frac{ρ}{2} h^{2}$ and $ν^{+} = ν + ρ h (x^{+})$ .

[F2]

Finite-budget augmented-Lagrangian multiplier method applies that update after an inner minimization.

[A1]

An exact inner routine computes this closed-form minimizer and returns success within its finite inner cap $M \geq 1$ ; arithmetic is exact and $ρ > 0$ .

Verification

1.1givenA1F1algebra

Substitution into [F1] gives $L_{ρ}^{E} (x, y; ν) = \frac{1}{2} x^{2} + \frac{1}{2} (1 + ρ) y^{2} + ν y .$ Its Hessian is diagonal with eigenvalues $1$ and $1 + ρ$ , so the Euclidean spectral condition number is $1 + ρ$ .

2.1A1F1F2step 1.1algebra∎

The gradient equations are $x = 0$ and $(1 + ρ) y + ν = 0$ ; the unique inner minimizer is $(0, - ν / (1 + ρ))$ . Its constraint value is $- ν / (1 + ρ)$ , so [F1] gives $ν^{+} = ν - ρ ν / (1 + ρ) = ν / (1 + ρ)$ . Its absolute contraction factor is $1 / (1 + ρ)$ , which decreases as $ρ$ increases while the condition number in step 1.1 increases.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Dimitri P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (standard reference, not scraped)