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Subdifferentials of proper extended-real convex functions
Definition
Let be proper and convex in the convention of Proper closed convex extended-real functions on . For and , define Write . If , set . When is finite-valued on a convex domain, agrees with Subgradients and the subdifferential of a convex function.
Depends on
- Proper closed convex extended-real functions on $\mathbb R^n$
- Subgradients and the subdifferential of a convex function
- The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$
- The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined
Used by
- Certified inexact proximal-gradient and FISTA updates Algorithm
- Coordinatewise minima of a separable convex composite are global minima Lemma
- Fenchel–Young inequality and its equality case Lemma
- One-step estimate for a certified inexact proximal call Lemma
- Optimality relations for exact two-block ADMM subproblems Lemma
- PDHG one-step primal-dual gap telescope Lemma
- The convex normal-cone operator is maximal monotone and its resolvent is projection Lemma
- Three-point inequality for the proximal-gradient model Lemma
- Two-block ADMM energy descent and objective gap estimates Lemma
- Existence, resolvent characterization, and firm nonexpansiveness of the proximal map Theorem
- Proximal bundle accumulation under vanishing aggregate-model error Theorem
- Strong convexity gives linear proximal-gradient and restarted-FISTA rates Theorem
Dependency tree · two levels
26 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, §2.3 (standard reference, not scraped)
- Mark Schmidt, Nicolas Le Roux, and Francis Bach, Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization, §3 and Appendix Lemma 2 (standard reference, not scraped)