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The function () = has ″ = >, so f is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2. The function () = has ″ =, so f is a convex function. It is strictly convex, even though the second derivative is not strictly positive at all points.
Strictly logarithmically convex if is strictly convex. Here we interpret as . Explicitly, f is logarithmically convex if and only if, for all x 1, x 2 ∈ X and all t ∈ [0, 1], the two following equivalent conditions hold:
A strongly convex function's second derivative is bounded away from zero. Following Boyd and Vandenberghe's book, we have: A twice continuously differentiable function is "strongly convex" if for all in the domain. The inequality is with respect to the positive semidefinite cone.
A Complex analytic space which admits a continuous strictly plurisubharmonic exhaustion function (i.e.strongly pseudoconvex) is Stein space. [4] Levi's problem remains unresolved in the following cases; Suppose that X is a singular Stein space, [note 22].
is a convex set. [2] The epigraphs of extended real-valued functions play a role in convex analysis that is analogous to the role played by graphs of real-valued function in real analysis. Specifically, the epigraph of an extended real-valued function provides geometric intuition that can be used to help formula or prove conjectures.
Every convex function is pseudoconvex, but the converse is not true. For example, the function f ( x ) = x + x 3 {\displaystyle f(x)=x+x^{3}} is pseudoconvex but not convex. Similarly, any pseudoconvex function is quasiconvex ; but the converse is not true, since the function f ( x ) = x 3 {\displaystyle f(x)=x^{3}} is quasiconvex but not ...
Update: Big Lots says it reached a deal in late December to keep hundreds of U.S. stores open. Big Lots is preparing to close all of its stores, the bankrupt discount retailer said Thursday. The ...
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation , Fenchel transformation , or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel ).