Solve Convex Function: Show yf(y^{-1}\textbf{x}) is Convex | Mathmos6

In summary, the conversation discusses how to show that the function yf(y^{-1}\textbf{x}) is convex on the interval x>0 when f is a convex function. The person asking for help has tried using the definition of convexity and manipulating the derivatives, but is unsure if their approach is correct. They are looking for a clearer and more efficient method to prove the convexity of the function.
  • #1
Mathmos6
81
0

Homework Statement


How do I show that if [tex]f\in C^2 \text{(}\mathbb{R}\text{)}[/tex] is convex then the function [tex]yf(y^{-1}\textbf{x})[/tex] is convex on (x,y):y>0?

Homework Equations



I know the standard definitions and whatnot about convexity, but I tried chugging through the algebra and didn't have any luck, can anyone show me a nice way to solve this?

Thanks!

-Mathmos6
 
Physics news on Phys.org
  • #3
Mark44 said:
Show us what you tried...

I figured [tex]yf(y^{-1}\textbf{x})=yf(\frac{x}{y},1)[/tex], so set [tex]f_x=\frac{\partial}{\partial{x}}f(\frac{x}{y},1), f_{yy}=\frac{\partial^2}{\partial{y^2}}f(\frac{x}{y},1)[/tex] and so on:

then we get

[tex]\frac{\partial{}}{\partial{x}}(yf(\frac{x}{y},1))=yf_x \Rightarrow \frac{\partial{}^2}{\partial{x^2}}=yf_{xx}[/tex]

and

[tex]\frac{\partial{}^2}{\partial{x}\partial{y}} (yf(\frac{x}{y},1))=f_x+yf_{xy}[/tex]

Also, W.R.T y:

[tex] \frac{\partial{}}{\partial{y}}=yf_y+f(\frac{x}{y},1) \Rightarrow \frac{\partial{}^2}{\partial{y}^2} = yf_{yy}+2f_y[/tex] - right?

Then looking at the Hessian, we know the first principal minor (=yfxx) is >=0 if y is, because f is convex so its corresponding first principal minor must also be >=0. With regards to the second principal minor though, i.e. the determinant of the Hessian, we get

[tex] \frac{\partial{}^2}{\partial{x}^2} \frac{\partial{}^2}{\partial{y}^2} - (\frac{\partial{}^2}{\partial{x}\partial{y}})^2=y^2(f_{xx}f_{yy}-f_{xy}^2)+2y(f_{xx}f_y-f_{xy}f_x)-f_x^2[/tex]

if my algebra is correct. We want to show this >=0 - the first term (the thing in the brackets multiplied by y2) is >=0 because it corresponds to the determinant of the Hessian for f - however, my concern is that I've gone wrong because something in the derivative of f(x/y,1) means this wouldn't work the same as for f(x,y) and so I'm not sure how to proceed... thanks :)
 

1. What is a convex function?

A convex function is a function that has a graph that curves upward, meaning that the function's value at any point on the graph is always less than or equal to the value at any point on the line segment between the two points. In other words, the function is always "bowl-shaped" and doesn't have any local minima or maxima.

2. How do you show that yf(y-1x) is convex?

To show that yf(y-1x) is convex, you can use the definition of convexity, which states that a function f is convex if and only if for any two points x1 and x2 in the function's domain and any value t between 0 and 1, the following inequality holds: f(tx1 + (1-t)x2) ≤ tf(x1) + (1-t)f(x2). You can also use the second derivative test to show convexity.

3. What is the significance of showing that yf(y-1x) is convex?

Showing that yf(y-1x) is convex is significant because convex functions have several useful properties, such as being easy to optimize and having a unique global minimum. This makes them useful in many fields, including economics, engineering, and statistics.

4. Can you provide an example of a convex function?

One example of a convex function is f(x) = x2, which has a graph that forms a parabola and is always increasing. Another example is f(x) = ex, which has a graph that curves upward and is always increasing.

5. Are there any other types of convex functions besides yf(y-1x)?

Yes, there are many other types of convex functions, such as f(x) = |x|, f(x) = log(x), and f(x) = x3. These functions all have graphs that are "bowl-shaped" and satisfy the definition of convexity.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
794
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
2
Replies
36
Views
3K
  • Calculus and Beyond Homework Help
Replies
19
Views
759
  • Topology and Analysis
Replies
24
Views
2K
Back
Top