DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

McCormick envelope for a bilinear term on a box

Definition

Let $x \in [l_{x}, u_{x}]$ and $y \in [l_{y}, u_{y}]$ , where all four endpoints are finite and ordered, and let $w$ be an additional real variable representing the product $x y$ . The McCormick polytope for this box is the set of $(x, y, w)$ satisfying the box bounds and $w \geq l_{x} y + l_{y} x - l_{x} l_{y}, w \geq u_{x} y + u_{y} x - u_{x} u_{y}$ $w \leq u_{x} y + l_{y} x - u_{x} l_{y}, w \leq l_{x} y + u_{y} x - l_{x} u_{y}$ The term McCormick envelope will mean this polytope. Its exact relation to the convex hull of the product graph is established by the local bilinear-hull result.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Arnold Neumaier, Complete Search in Continuous Global Optimization and Constraint Satisfaction, §16 (standard reference, not scraped)
A. Tsoukalas and A. Mitsos, Multivariate McCormick relaxations, §5 (standard reference, not scraped)