Proving Convexity of a Function with Directional Derivative

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In summary, the conversation discusses the definition of convexity and the proof of a proposition related to it. The issue raised in the conversation is about the use of directional derivatives and the result obtained from it. The expert summarizer notes that there is no need for a unit vector and the crucial point is ##\mu \to 0##, as explained in a referenced source.
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I'm reading book and there's proposition with convex function

Function f is convex if and only if for all x,y
[tex](*)\quad f(x)-f(y)\ge\nabla f(y)^T(x-y)[/tex]

It's proven in this way: From definition of convexity

[tex]f(\lambda x+(1-\lambda)x)\le \lambda f(x)+(1-\lambda)f(y)[/tex]

we have

[tex]\frac{f(y+\lambda(x-y))-f(y)}{\lambda}\le f(x)-f(y)[/tex]

Setting [tex]\lambda\to0[/tex] we have (*).

-------------------------------------------------

My problem is in last sentece. I understand formula on left-hand side as directional derivative. But in definition of directional derivative is needed to (x-y) be an unit vector. It is not. So it is not a directional derivative. If I define

[tex]\lambda:=\frac{\mu}{\Vert x-y\Vert}[/tex]

I have directional derivative on left hand side

[tex]\frac{f(y+\mu\frac{(x-y)}{\Vert x-y\Vert})-f(y)}{\mu}\le\frac{f(x)-f(y)}{\Vert x-y\Vert}[/tex]

But in this way I don't obtain result (*) but I obtain this

[tex]\nabla f(y)^T(x-y)\le\frac{f(x)-f(y)}{\Vert x-y\Vert}[/tex]
 
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1. What is the definition of a convex function?

A convex function is a function whose graph lies above any chord connecting two points on the graph. Mathematically, it means that for any two points on the graph, the value of the function at the midpoint of the chord is less than or equal to the average of the function values at the two points.

2. How can I determine if a function is convex using directional derivatives?

A function is convex if and only if its directional derivatives are non-decreasing in all directions. This means that as you move in any direction on the graph, the slope of the tangent line at any point on the graph must be equal to or greater than the slope of the tangent line at any point in the same direction.

3. Can I use directional derivatives to prove convexity for any type of function?

Yes, directional derivatives can be used to prove convexity for any type of function, as long as the function is differentiable at the point in question.

4. What is the significance of proving convexity of a function?

Proving convexity of a function is important because it guarantees that the function has a unique global minimum. This makes it useful in optimization problems, as it helps determine the most efficient solution.

5. Are there any other methods for proving convexity of a function?

Yes, there are other methods for proving convexity of a function, such as using the second derivative test or checking for positive-definiteness of the Hessian matrix. However, using directional derivatives is often the simplest and most straightforward approach.

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