# 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

## 1 Answer

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.

• So sir convexity doesn't depend upon the parameter values? if I multiply the constraint by -1, it becomes Rk - Wk*(B/N)*log2(1+ (Pk * Hk)/(Wk * sigma)) <= 0, it needs to be convex. But sir if I find the hessian of the LHS and substitute the values of the parameters as in the question, I am getting the eigen values of the hessian as negative. but sir, for it to be convex the hessian should be positive semi definite. – cody Oct 17 '13 at 18:05
• 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
• Sir I have written the following code to solve the above problem in YALMIP. [link][1] [1]: drive.google.com/file/d/0B39N8ztL-yN3RWEzNUJnblNIaDg/… – cody Oct 24 '13 at 9:27
• But sir Matlab is giving me an error saying.. EXIT: Iterates diverging; problem might be unbounded. info: 'Numerical problems (IPOPT)' problem: 4 – cody Oct 24 '13 at 9:27
• Have I not written the code properly Or is it because of the values of the parameters(can they be wrong, I'm not sure of them). Because the problem is convex and IPOPT is unable to solve it. Or should I use any other solver for this particular problem. – cody Oct 24 '13 at 9:28