Extremum Problems with Inequalities as Subsidiary Conditions

By F. John. 2014

829 citations recorded by the source corpus.

Subfields: Aerospace Engineering, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/b3d8ea03879afd37d785dfe4dd9234d62cd2c0c9

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In Aerospace Engineering

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Bandit Algorithms (2020, background): “John’s theorem is due to John [1948]. The duality mentioned in the text was proved by Silvey and Sibson [1972].”
On the Theory of Policy Gradient Methods: Optimality, Approximation, and Distribution Shift (2019, background): “There always exists a state-action distribution ν, which is supported on at most d2 state-action pairs and which can be constructed only with knowledge of Φ (without knowledge of the MDP or d⋆), such that: κ ≤ d. Proof: The distribution can be found through constructing the minimal volume ellipsoid ”
The Dual Orlicz–Minkowski Problem (2017, background): “, [24, 31, 34, 39, 40, 48, 54]), the Lp Minkowski problem for p ≥ 1 is closely related to optimization problems involving the Lp mixed volume of convex bodies; this can be seen intuitively from the equality characterization of the Lp Minkowski inequality for the Lp mixed volume.”
Relatively Smooth Convex Optimization by First-Order Methods, and Applications (2016, background): “geometry, D-optimal design arises as a Lagrangian dual problem of the minimum volume covering ellipsoid (MVCE) problem, which dates back at least 60 years to [9], see Todd [21] for a modern treatment.”
On the Monge-Amp\`ere equation (2018, methodology): “The intuition of Caffarelli was to use the so-called John’s Lemma [42]: Lemma 3.”
The Binet-Legendre Metric in Finsler Geometry (2011, background): “Arguing as in [29] one can slightly improve the inequality (2.”
Learning search spaces for Bayesian optimization: Another view of hyperparameter transfer learning (2019, methodology): “More formally, Using the classical Löwner-John formulation [21], the lowest volume ellipsoid covering all points { x } Tt =1 is the solution to the following problem (see Section 8.4 in [5]): where the T norm constraints enforce x ∈ ˆ X ( θ ) , while the minimized objective is a strictly increasing…”
Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths (2011, neutral): “onentially decaying factor in (3.13), it is still possible to obtain a uniform bound, but one needs to be a little bit more careful. Our main tool is the fact that, as a consequence of John’s theorem [2, 26], it is possible to perform an orthogonal change of coordinates for ϕ(that depends on Y) and ”
On Pólya’s Inequality for Torsional Rigidity and First Dirichlet Eigenvalue (2016, neutral): “By the John’s ellipsoid Theorem ([9]), there exists an ellipsoid Υ with centre c such that Υ ⊂ Ω ⊂ c+m(Υ− c).”
On the Euler-Lagrange equation of a functional by Pólya and Szegö (2016, methodology): “In particular, Crasta, Fragalà and Gazzola,using the John ellipsoid theorem [12], proved in [3] that the functional F has a minimum in K3.”
Quantitative stability for the Brunn-Minkowski inequality (2014, neutral): “+1 ∪ t′′ BΣ +(1−t′′B )y , i= 1,...,n−1. Then (i) Hi−1(Σ i) ≥ cn, Hi−1(Σ i ∩C¯1) Hi−1(Σ i) ≥ 1 −δζ/2, ∀i= 2,...,n; (ii) Hi−1 Σ′ i ∩C¯1 Hi−1 (Σ′ i) ≥ 1 −δζ/2 ∀i= 2,...,n−1. Then, thanks to John’s Lemma [J], up to an aﬃne transformation of the form (4.41) we can assume that Br ⊂ Ω ⊂ B(n−1)r, 1/Cn <r”