Convexity is a mathematical concept that describes a property of a function or shape that is similar to a bowl or a cup. It means that for any two points on the function, the line connecting them will lie above the function.
To prove that a function is convex, you need to show that the line connecting any two points on the function lies above the function. This can be done by taking the second derivative of the function and showing that it is always positive.
Proving convexity is important because it guarantees that a function has only one minimum point, making it easier to find the optimal solution to a problem. It also ensures that the function is well-behaved and has certain properties that can be used in mathematical calculations.
One example of a convex function is the squared norm function, f(x)=||x||^2. This function is convex because the line connecting any two points on the function will always lie above the function.
The convexity of a function is essential for optimization because it ensures that the function has a unique minimum point. This allows for efficient and reliable optimization techniques to be used, such as gradient descent, to find the optimal solution to a problem.