Alphabeta MathHow 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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Proximal-gradient mapping and projected-gradient mapping
Definition
Let be continuously differentiable with -Lipschitz gradient for some , and let be proper, closed and convex. Define The proximal theorem Existence, resolvent characterization, and firm nonexpansiveness of the proximal map ensures the point is uniquely defined. For a nonempty closed convex set and , the projected-gradient update is where denotes the unique nearest-point projection; this equality follows from The proximal map of an indicator is Euclidean projection.
Depends on
- Proper closed convex extended-real functions on $\mathbb R^n$
- Proximal mapping
- Extended-real indicator of a set
- Existence, resolvent characterization, and firm nonexpansiveness of the proximal map
- The proximal map of an indicator is Euclidean projection
- The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$
Used by
Dependency tree · two levels
31 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
- Neal Parikh and Stephen Boyd, Proximal Algorithms, §4.2 (standard reference, not scraped)
- Amir Beck, First-Order Methods in Optimization, SIAM, 2017, Chapter 10 (standard reference, not scraped)