Convex Analysis¶
Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press.
Cited by¶
6 citations across 6 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Constraint
- … and its 1951 independent rediscovery and extension by Kuhn and Tucker established the KKT necessary conditions for constrained nonlinear optimization, generalizing Lagrange's framework to inequality constraints and remaining the foundational result of modern optimization theory; Rockafellar's 1970 Convex Analysis
This sourceFoundational treatise on convex analysis: formalizes curvature direction (convex vs. concave) as the primary geometric distinction in functional response and develops the duality framework underlying modern convex theory.
- … and its 1951 independent rediscovery and extension by Kuhn and Tucker established the KKT necessary conditions for constrained nonlinear optimization, generalizing Lagrange's framework to inequality constraints and remaining the foundational result of modern optimization theory; Rockafellar's 1970 Convex Analysis
- Convexity
- Convexity unlocks several reusable inference patterns, each stated in terms of mixtures and chords rather than any substrate. Jensen's inequality — for convex \(f\), the expectation of \(f(X)\) is at least \(f\) of the expectation — is the single source of the AM–GM inequality, the entropy bound, log-sum inequalities in information theory, and risk-premium calculations in finance.
This sourceCanonical treatise on convex sets and functions — Jensen's inequality, the separating-hyperplane theorem, Carathéodory's theorem, and duality.
Supported in partVerified against the publisher's abstract
The abstract shows the book treats convex sets, systems of inequalities, duality and economic applications — the claim's frame only; it names neither the separating-hyperplane theorem, Farkas's lemma, nor the logic or finance attributions.
“The topics treated in this volume include: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle- functions. This book has firmly established a new and vital area not only for pure mathematics but also …”
- Convexity unlocks several reusable inference patterns, each stated in terms of mixtures and chords rather than any substrate. Jensen's inequality — for convex \(f\), the expectation of \(f(X)\) is at least \(f\) of the expectation — is the single source of the AM–GM inequality, the entropy bound, log-sum inequalities in information theory, and risk-premium calculations in finance.
Domain-specific¶
- Interior
- In the theory of convex sets, the relative interior of a convex set (its interior relative to its affine hull) is always non-empty and supports the dimensional arguments in convex duality and separation theorems
This sourceRockafellar's Convex Analysis, stating that a non-empty convex set has a non-empty relative interior, in a book covering convex-set structure and duality.
Supported in partVerified against the work's full text
“has a non-empty [relative interior]. §2. CONVEX SETS AND CONES”
- In the theory of convex sets, the relative interior of a convex set (its interior relative to its affine hull) is always non-empty and supports the dimensional arguments in convex duality and separation theorems
- Proper Convex Function
- Strong duality
- Subderivative
Verification¶
Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.
Does it back the claim? Read against the text for 2 of 6 citations: 2 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
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Links previously used in the corpus¶
Before the registry existed this work was also linked 2 other ways.
Registry ID ref:510501566eaf · see in the full table