Convex Functions: Constructions, Characterizations and Counterexamples

By Jonathan Michael Borwein, J. Vanderwerff. 2010

371 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/c7281e3c79123d40c3709e0043624636e0114c31

How researchers see this paper

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

In Applied Mathematics

Citations in the corpus

Strongly Convex Functions, Moreau Envelopes, and the Generic Nature of Convex Functions with Strong Minimizers (2015, background): “We refer the reader to [7, 9, 11, 27] for further details on epi-convergence, e.”
Functional Versions of Lp-Affine Surface Area and Entropy Inequalities (2014, neutral): “…when ψ is neither smooth nor strictly convex, the gradient of ψ , denoted by ∇ ψ , exists almost everywhere by Rademacher’s theorem (see, e.g., [10]), and a theorem of Alexandrov [1] and Busemann and Feller [11] guarantees the existence of the Hessian, denoted by ∇ 2 ψ , almost everywhere in Ω…”
A Primer on Monotone Operator Methods (2015, background): “Standard references on convex optimization and convex analysis (which contain far more than the reader needs) include [101, 14, 18, 86, 15, 110, 12, 16, 11, 13].”
Forward-Backward and Tseng’s Type Penalty Schemes for Monotone Inclusion Problems (2013, background): “For the readers convenience we present first some notations which are used throughout the paper (see [4, 7, 8, 16, 22, 25]).”
Inertial Douglas-Rachford splitting for monotone inclusion problems (2014, methodology): “ the inertial Douglas-Rachford primal-dual algorithm can be implemented when solving a primal-dual pair of convex optimization problems. We recall rst some notations used in the variational case, see [6,8,9,24,35,37]. For a function f : H!R, where R := R [f1g is the extended real line, we denote by ”
The spatial distribution in infinite dimensional spaces and related quantiles and depths (2014, methodology): “5 in [3], we have Sx(v) = E{SGNx(v)−X} = v.”
A Dynamical System Associated with the Fixed Points Set of a Nonexpansive Operator (2014, background): “continuous version of the forward-backward algorithm. For readers convenience let us recall some standard notions and results in monotone operator theory which will be used in the following (see also [7,9,23]). For an arbitrary set-valued operator A : H ⇒ H we denote by GrA = {(x,u) ∈ H × H : u ∈ Ax”
Some new inequalities of Hermite-Hadamard type for s-convex functions with applications (2017, neutral): “A large number of important properties and inequalities have been established for the class of convex (concave) functions since the convexity (concavity) was introduced more than a hundred years ago [1-21].”
Extension of Convex Function (2012, background): “ex domain, convex hull, positive deﬁ- nite Hessian, extension 1 Introduction Convex functions appear in many important problems in pure and applied mathematics. In many literatures on convex analysis [2, 8, 9], convex functions are usually deﬁned only on convex domains. Moreover, convex functions ar”