Convex functions

  • How do you know if a function is convex?

    1.
    If you know calculus, take the second derivative.
    It is a well-known fact that if the second derivative f (x) is ≥ 0 for all x in an interval I, then f is convex on I.
    On the other hand, if f(x) ≤ 0 for all x ∈ I, then f is concave on I..

  • How do you know if a function is strictly convex?

    If f is twice differentiable, it is enough to check that the second derivative is non negative.
    Check the Hessian matrix of the function.
    If the matrix is: Positive-definite then your function is strictly convex..

  • How is a convex function shaped?

    A function on a graph is convex if a line segment drawn through any two points on the line of the function never lies below the curved line segment.
    I.e., basically, a convex function has its curve opening upward like a cup..

  • What are the conditions for a function to be convex?

    A function f is convex, if its Hessian is everywhere positive semi-definite.
    This allows us to test whether a given function is convex.
    If the Hessian of a function is everywhere positive definite, then the function is strictly convex..

  • What convex function means?

    In simple terms, a convex function refers to a function whose graph is shaped like a cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap ..

  • What is a convex shaped function?

    A function on a graph is convex if a line segment drawn through any two points on the line of the function never lies below the curved line segment.
    I.e., basically, a convex function has its curve opening upward like a cup..

  • Which functions are convex?

    A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval..

  • f is both concave and convex iff for any a, b ∈ RN and any θ ∈ (0,1), f(θa + (1 − θ)b) = θf(a) + (1 − θ)f(b).
    A function f is affine iff there is a 1 \xd7 N matrix A and a number y∗ ∈ R such that for all x ∈ C, f(x) = Ax + y∗. f is linear if it is affine with y∗ = 0.
Convex Function Summary A function is convex if it lies above its tangent line at every point. More formally, a function f(x) is convex if for any x1, x2 in its domain and any t between 0 and 1, we have f(tx1+(1−t)x2)≤tf(x1)+(1−t)f(x2).
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph is a convex Wikipedia
In simple terms, a convex function refers to a function whose graph is shaped like a cup (or a straight line like a linear function), while a concave function'  Logarithmically convexK-convex functionConcave functionJensen's inequality

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