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Intuitively, a strongly-convex function is a function that grows as fast as a quadratic function. [11] A strongly convex function is also strictly convex, but not vice versa. If a one-dimensional function f {\displaystyle f} is twice continuously differentiable and the domain is the real line, then we can characterize it as follows:
The Banach space (X, ǁ ⋅ ǁ) is a strictly convex space (i.e., the boundary of the unit ball B contains no line segments) if and only if δ(2) = 1, i.e., if only antipodal points (of the form x and y = −x) of the unit sphere can have distance equal to 2. When X is uniformly convex, it admits an equivalent norm with power type modulus of ...
Packet Tracer is a cross-platform visual simulation tool designed by Cisco Systems that allows users to create network topologies and imitate modern computer networks.The software allows users to simulate the configuration of Cisco routers and switches using a simulated command line interface.
Convex analysis includes not only the study of convex subsets of Euclidean spaces but also the study of convex functions on abstract spaces. Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets , often with applications in convex minimization , a subdomain of optimization theory .
We are also given differentiable convex function :, -strongly convex with respect to the given norm. This is called the distance-generating function , and its gradient ∇ h : R n → R n {\displaystyle \nabla h\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}} is known as the mirror map .
A function defined on a rectangle (top figure, in red), and its trace (bottom figure, in red). In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space.
assert that y not in S(K,-ε).Closely related to the problems on convex sets is the following problem on a compact convex set K and a convex function f: R n → R given by an approximate value oracle: Weak constrained convex function minimization (WCCFM): given a rational ε>0, find a vector in S(K,ε) such that f(y) ≤ f(x) + ε for all x in ...
It would be great to add a section with the definition of a strongly convex function; or, make a page for "strongly convex" (since it appears that it doesn't yet exist) and then link to that page. 71.130.221.31 02:46, 2 February 2009 (UTC) Well, I'd like to know what a "strongly convex" function is?