Minimax Theorems.

By K. Fan. 1953

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In Scaling & Learning Theory

Citations in the corpus

Adversarial Example Games (2020, methodology): “We finally use Fan’s theorem [28] after showing that (fc, pg) 7→ φλ(fc, pg) is convex-concave (by convexity of ` and linearity of pg 7→ Epg ) and that ∆ is a compact convex set.”
Soliton solutions for quasilinear Schrödinger equations with critical growth (2010, methodology): “We would like to cite papers by Noussair, Swanson and Yang [31], Miyagaki [29], García and Peral [16], Benci and Cerami [4] and the book of Willem [41].”
Large deviations for Markovian nonlinear Hawkes processes (2011, methodology): “…upper semicontinuous in (ˆ λ ˆ π, ˆ π ) and convex in f , it is suﬃcient to prove the compactness of R to apply Ky Fan’s minmax theorem (see Fan [7]). level Indeed, Jo´o developed some set method and proved that it is suﬃcient to show the compactness of the level set (see Jo´o [13] and Frenk…”
Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials (2007, background): “[26]), it is not difficult to check that ∫”
An optimal partition problem related to nonlinear eigenvalues (2003, unknown): “By the quantitative deformation lemma (see, for instance, [17]) we derive the existence of a continuous map η : (H1 0 (Ω)) k → (H1 0 (Ω))k and of a constant ν > 0 such that: (i) η(U) = U for all U such that ‖U − U c‖ > 4δ or |I(U)− c| > 2ν; (ii) I(η(U)) ≤ I(U) for all U ∈ (H1 0 (Ω))k; (iii) U ∈ B2δ(”
Existence results to elliptic systems with nonstandard growth conditions (2004, methodology): “Applying Lemmas 2 a we conclude that J has at least one nontrivial critical poi nt [13,17,18].”
Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space (2011, methodology): “2 is based on Theorem A, the mountain-pass theorem without the Palais–Smale condition [33] and the Ekeland’s variational principle [36], which were also used in [5,21].”
BEST RESPONSE DYNAMICS FOR CONTINUOUS ZERO{SUM GAMES (2005, background): “Under the above assumptions on U , the minmax theorem ([21], [9]) applies and the game is known to have a value.”
Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent (2017, methodology): “4, there exits a Palais–Smale sequence (un)n∈N by the mountain pass theorem (see for example [1, 28, 30, 36]) at the energy level c0 defined by (2.”
Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions (2004, methodology): “We used variational methods such as mountain pass theorem without Palais-Smale condition (see [14]) to obtain a positive ground-state solution.”
Existence of groundstates for a class of nonlinear Choquard equations (2012, methodology): “…set of paths is deﬁned as Classically, in order to show that b is a critical level of the functional I , one constructs a Palais–Smale sequence at the level b , that is, a sequence Then one proves that the sequence ( u n ) n ∈ N converges up to translations and extraction of a subsequence [37, 43].”
$\propto$SVM for learning with label proportions (2013, background): “Because the objective function of (14) is convex in μ, and concave in α, it is equivalent to (Fan, 1953),”
Existence of Groundstates for a Class of Nonlinear Choquard Equations in the Plane (2016, methodology): “2, the mountain pass level b is not trivial, we can thus apply the minimax principle ([24], Theorem 2.”
Existence of solutions for p(x)-Laplacian problems on a bounded domain (2005, background): “The mountain pass theorem (see [21]) guarantees that J s a nontrivial critical point u.”