Convex Functions

Flexing and Convexing

When you hear the word convex in your proofs, please listen carefully as mathematicians want to tell you something.

Convexity has been there for a long time now, but in this post we are making a beginning into the world of CONVEX OPTIMIZATION, which begins by understanding what convex functions means.

First of all, lets see what CONVEX SET means

and

A few examples of convex sets are

Figures in first row are convex sets while second row are not convex sets, which we saw by constructing a chord which lies outside the figure.

Now we can move ahead with convex functions.

and Graphically just like convex sets, we construct a chord between any two points(say a and b), then pick up a point in between a and b say c, then the chord value for c is greater than the functional value of c, we can see by some figures

Now, there exists opposite of convex functions which is Concave functions, where chord value is less than the functional value of c.

Testing for convexity for single variable function

For a twice-differentiable function f, if the second derivative, f ‘’(x), is positive (or, if the acceleration is positive), then the graph is convex (or concave upward); if the second derivative is negative, then the graph is concave (or concave downward).

Resources:

https://cpb-us-w2.wpmucdn.com/sites.gatech.edu/dist/2/436/files/2017/07/04-convex-funcs.pdf, https://web.mit.edu/~jadbabai/www/EE605/lectures/functions.pdf, https://people.math.aau.dk/~jjohnsen/Teaching/Notes/convexity.pdf