# Convexity Check

I have the following optimization problem and I am trying to check for its convexity.

link As per the definition of convexity, "a continuous twice differentiable function is convex ON a convex set, iff the hessian is positive semi definite on the interior of the convex set." The feasible region where I am trying to check for its convexity its convexity is defined by..

B, N, sigma are constants

B = 20000

N = 50

sigma = 3.7678e-17

w, p are variables(vectors of K elements)

w can vary from 1 to 50

p, 0 to 46 in dbm(unit of power..)

I have to convert in into watt before using it

p in watt = (10^(p in dbm/10))/1000.

The hessian comes out to be indefinite for MANY of the feasible points..But I am not sure if I am right. Is the problem CONVEX, can any one check please..?

• As in your other post, please write out your nonlinear program in LaTeX in the original post rather than post a link to an external site. – Geoff Oxberry Oct 18 '13 at 18:05

For the problem to be convex, the nonlinear expression in your first constraint has to be concave. Indeed, this is the case at it essentially is a negated relative entropy (known to be convex), $w\log(1+\frac{p}{w}) = -w\log(\frac{w}{p+w})=-relentr(w,p+w)$. Alternatively, we note that it is the perspective of $f(p)=\log(1+p)$, and it thus follows that $wf(p/w)$ is concave.
• Exactly, $R_k - \log(...)$ must be convex, which it is since the logarithm is concave. The set satisfying $f(x) \leq 0$ is convex if $f(x)$ is convex. If you multiply by $-1$ you get $-f(x) \geq 0$ and you get the condition that $-f(x)$ must be concave, i.e., $f(x)$ convex. – Johan Löfberg Oct 17 '13 at 19:19