Geometric Functional Analysis and Its Applications

By R. Holmes. 1975

Subfields: Mathematical Optimization, Pure Mathematics

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Optimization

In Pure Mathematics

Citations in the corpus

Frank-Wolfe Splitting via Augmented Lagrangian Method (2018, background): “It is a particular case of constraint qualifications (Holmes, 1975; Gowda and Teboulle, 1990).”
Localized Tight Frames on Spheres (2006, background): “By the Krein– Rutman theorem [9], there exists a positive linear functional Ψ̃X that extends ΨX to all R|X|.”
ESSENTIAL SMOOTHNESS, ESSENTIAL STRICT CONVEXITY, AND LEGENDRE FUNCTIONS IN BANACH SPACES (2001, background): “Facts Most of the following results are known, and their proofs can be pieced together from various sources such as [1, 3, 27, 29, 36, 37, 43].”
On the existence and stability of approximate solutions of perturbed vector equilibrium problems (2007, background): “In [15], the notion of rotund set is introduced: a nonempty convex set A ⊂ X is said to be rotund when its boundary does not contain line segments.”
Regularity conditions via generalized interiority notions in convex optimization: New achievements and their relation to some classical statements (2009, background): “We refer the reader to [4,6,25,26,36,40,44,47] and to the references therein for more properties and examples regarding the above considered generalized interiority notions.”
Connes' embedding conjecture and sums of hermitian squares (2006, methodology): “We avoid this terminology and instead use the concept of algebraic interior (or core) points [ Hol , p. 7, §2C].”
Convex Functions: Constructions, Characterizations and Counterexamples (2010, background): “As in [260], let C be a closed bounded convex subset of a normed space E and let 0 = x0 ∈ E.”
Convex Analysis And Nonlinear Optimization (2000, unknown): “The Open mapping theorem (Exercise 9) is another fundamental tool of functional analysis [98].”
Existence and Stability of Solutions for Generalized Ky Fan Inequality Problems with Trifunctions (2010, background): “is upper semicontinuous and, for each (x, y) ∈ E × L ,t he setF(x,y) is a nonempty closed convex subset of E × L. By the Kakutani-Fan-Glicksberg fixed-point theorem (see [ 13 ]), there exists a point ( ¯ y)∈ E × L such that ( ¯ x, ¯ y)∈ F( ¯ x, ¯ y), that is,”
Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces (2011, unknown): “It is well known (see for example page 170 in [20]) that G is Fréchet differentiable, with G ′(v) = v |v|p−2 ‖v‖1−p Δp ∈ Lμ(I1 × I2), with q such that 1/q + 1/p = 1.”
ON DUALITY THEORY OF CONIC LINEAR PROBLEMS (2001, background): “, [11]) for a theory of locally convex topological vector spaces.”
The failure of rational dilation on a triply connected domain (2003, background): “We have λ ( I ) > 0, as otherwise, from Lemma 5.1, λ ( H ( z ) H ( w ) ∗ ) = 0 for all H and hence λ = 0 (see, for example, Holmes [23] Given f analytic in a neighborhood of X , consider the mapping M f : H 2 ( X ) → H 2 ( X ) deﬁned by multiplication by f so that M f g = fg .”