Any midpoint convex function that satisfies reasonable further conditions is convex. Because convex functions are bounded on subintervals, it follows that convex functions are continuous.ELSEVIERPure and Applied Mathematics
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We provide a second-order stochastic differential equation (SDE), which characterizes the continuous-time dynamics of accelerated stochastic mirror descent (ASMD) for strongly convex functions. This SDE plays a central role in designing new discrete-time ASMD algorithms via numerical discretization and ...
of roughly convex functions in the sense of Kltzler (it suffices to consider points with a distance equal to the roughness degree) is verified, a sharpened y-convexity condition (which is satisfied by Favard's "fonction penetrante") is introduced, and some continuous counterexamples are glven...
Melzer, D. : ‘On the expressibility of piecewise-linear continuous functions as the difference of two piecewise-linear convex functions’, Math. Program. Stud. 29 (1986), 118–134.D. Melzer. On the expressibility of piecewise-linear continuous functions as the difference of two piecewise-...
Integrally convex functions serve as a common framework for discrete convex functions. Indeed, separable convex, L-convex, L-convex, M-convex, M-convex, L-convex, and M-convex functions are known to be integrally convex [40]. Multimodular functions [23] are also integrally convex, as pointed...
In the geometries of stratified groups, we show that H-convex functions locally bounded from above are locally Lipschitz continuous and that the class of v-convex functions exactly corresponds to the class of upper semicontinuous H-convex functions. As a consequence, v-convex functions are locally...
Continuity of Convex Functions Proposition 1.3.11: If f : ℜn → ℜ is convex, then it is continuous. More generally, if f : ℜn → (−∞, ∞] is a proper convex function, then f , restricted to dom(f ), is continuous over the relative interior of dom(f ). 476 Chap. ...
(1) Are such characterizations valid for lower semicontinuous (l.s.c.) functions? (2) Which subdifferentials can be used for characterizing such properties? Question (1) is not without importance: in optimization it is useful to use l.s.c. functions, in particular to substitute a constraint...
[ 43 ]. the inequalities, introduced by c. hermite and j. hadamard for convex functions, are of considerable significance in the literature. these inequalities state that, if \(\digamma :i\rightarrow \mathbb{r}\) is a convex function on the interval i of real numbers and \(a,b\in i...