ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

Norm-bounded robust least squares as an SOCP

Example

Let $m, n \geq 1$ , $A \in R^{m \times n}$ , $b \in R^{m}$ , and $ρ \geq 0$ . Suppose an adversary may perturb $A$ by any $Δ \in R^{m \times n}$ with $∥ Δ ∥_{F} \leq ρ$ . The robust residual objective is $\sup_{∥ Δ ∥_{F} \leq ρ} ∥ (A + Δ) x - b ∥_{2}$ .

Facts & Assumptions

Given: The dimensions, data, and perturbation set above.

[F1]

$∥ Δ x ∥_{2} \leq ∥ Δ ∥_{F} ∥ x ∥_{2}$ and the Euclidean triangle inequality (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F2]

A constraint $∥ v ∥_{2} \leq t$ is Lorentz-cone membership $(t, v) \in L$ (Conic, LP, and convex QP primal-dual models).

Verification

1.1F1givenconstruct

For every admissible $Δ$ , [F1] gives $∥ (A + Δ) x - b ∥_{2} \leq ∥ A x - b ∥_{2} + ρ ∥ x ∥_{2}$ . If $x \neq 0$ and $r = A x - b \neq 0$ , choose $u = r / ∥ r ∥_{2}$ and $Δ = ρ u x^{T} / ∥ x ∥_{2}$ ; then $∥ Δ ∥_{F} = ρ$ and $Δ x = ρ ∥ x ∥_{2} u$ , attaining the upper bound. If $r = 0$ , any unit $u$ gives the same supremum; if $x = 0$ , the perturbation term vanishes.

2.1F2step 1.1algebra∎

Therefore the robust objective equals $∥ A x - b ∥_{2} + ρ ∥ x ∥_{2}$ . Introducing epigraph variables $t, u$ gives $\min t + ρ u$ subject to $(t, A x - b) \in L_{m + 1}$ and $(u, x) \in L_{n + 1}$ by [F2], which is an SOCP.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

Stephen Boyd and Lieven Vandenberghe, Convex Optimization (standard reference, not scraped)
Hamza Fawzi, Topics in Convex Optimisation, Lecture 16 (standard reference, not scraped)