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A function (in black) is convex if and only if the region above its graph (in green) is a convex set. A graph of the bivariate convex function x 2 + xy + y 2. Convex vs. Not convex
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
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.
A function is convex if and only if its epigraph, the region (in green) above its graph (in blue), is a convex set.. Let S be a vector space or an affine space over the real numbers, or, more generally, over some ordered field (this includes Euclidean spaces, which are affine spaces).
This set is convex because is convex, the sublevel sets of convex functions are convex, affine sets are convex, and the intersection of convex sets is convex. [7]: chpt.2 Many optimization problems can be equivalently formulated in this standard form.
This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are convex. [1] Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x 2 /2) which is log-concave since log f(x) = −x ...
A convex curve (black) forms a connected subset of the boundary of a convex set (blue), and has a supporting line (red) through each of its points. A parabola, a convex curve that is the graph of the convex function () = In geometry, a convex curve is a plane curve that has a supporting line through each of its points.
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).