Contents
What makes a function 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. More generally, a function is convex on an interval if for any two points and in and any where , (Rudin 1976, p. 101; cf.
What is the convex concave rule?
The concave-convex rule states that if a concave surface moves on a convex surface, roll and slide must occur in the same direction, and if a convex surface moves on a concave surface, roll and slide occurs in opposite directions.
Is Norm a convex function?
Every norm is a convex function, by the triangle inequality and positive homogeneity. The spectral radius of a nonnegative matrix is a convex function of its diagonal elements.
What is the concave rule?
How can you tell if a function is convex?
For single variable functions, you can check the second derivative. If it is positive then the function is convex. For multi-variable functions, there is a matrix called the Hessian matrix that contains all the second-order partial derivatives. The Hessian matrix being positive definite is the condition for convexity here.
When do you know if a matrix is convex?
Positive-definite then your function is strictly convex. Positive semi-definite then your function is convex. A matrix is positive definite when all the eigenvalues are positive and semi-definite if all the eigenvalues are positive or zero-valued.
When is a function convex it’s Hessian positive semidefinite?
For single variable functions, you can check the second derivative. If it is positive then the function is convex. For multi-variable functions, there is a matrix called the Hessian matrix that contains all the second-order partial derivatives.
Can a function with a non positive second derivative be convex?
Such a function can still be convex; for example Somebody changed my answer, saying that a function with a non- positive second derivative is convex.