I am trying to show the convexity of functions which include exponential variables with different powers.
My question is, for quadratic functions($x^TQx$) it is easier to show as showing Q is nonnegative is enough to prove. Or we know linear functions are convex. But for these problems it is different. And finding Hessian of each function can be very time consuming task. What I want to ask is, do I need to find all of the expressions'(the objective functions and the constraints) Hessians to prove they are all convex? Or is there any other shorter way to do that?
Sample problems:
Sample Problem 1
Minimize $|2x_1+3x_2+x_3|+x_1^2+x_2^2+x_3^2+\sqrt{2x_1^2+4x_1x_2+7x_2^2+10x_2+6}$
s.t. $\frac{x_1^2+1}{x_2}+2x_1^2+5x_2^2+10x_3^2+4x_1x_2+2x_1x_3+2x_2x_3 \leq7$
Sample Problem 2
Minimize $\sqrt{2x_1^2+3x_2^2+x_3^2+4x_1x_2+7}+(x_1^2+x_2^2+x_3^2)^2$
s.t. $\frac{(x_1+x_2)^2}{x_3+1}+x_1^8\leq7$
$x_1^2+x_2^2+4x_3^2+2x_1x_2+2x_1x_3+2x_2x_3\leq10$
$x_1,x_2,x_3\geq0$
I know that sum of two strictly convex is convex. Is there any similar properties to make these proofs easier?
Best Answer
I found the answer. Using sum of convex functions are convex property is enough and very useful here. That way, I was able to split the problem into subproblems and solved it.
For instance, take the first sample. $|2x_1+3x_2+x_3|$ here, absolute value of a convex function is always convex. $x_1^2+x_2^2+x_3^2$ here, as this is a quadratic function, it can be written in the form $x^TQx$ where $Q$ matrix is the identity matrix which is positive definite. Therefore, it is convex, too. For the last expression of the objective function it is a bit trickier. $\sqrt{2x_1^2+4x_1x_2+7x_2^2+10x_2+6}$ it can be written in the following form as inside of the square root is quadratic: $\sqrt{x^TQx+k}$. Here Q is positive definite as well, I am not going to get into the proof of that one, but you can find it easily with a simple search. And, we are able to prove that the objective function is convex. I'll stop here but you should do that for the constraints to show that feasible region consists of a convex set.