How to check the convexity of the objective function?

Hi Everyone, I have the following objective function: Max 1-exp(- SNR_threshold/average SNR) and subject to power constraints,0>p< P_max. How can I check if it convex problem or not?

Answer 1

[Kunal Kandhari]

Hi Alexa,

Since, If the second derivative of a function is always positive or non-negative over an interval, then the function is convex over that interval so you can use the "diff" function in MATLAB to calculate the derivatives and check if f'' >= 0

You can read about the usage of "diff" function from the following documentation:

https://in.mathworks.com/help/matlab/ref/diff.html

[Alexa]

@Kunal Kandhari Thank you so much! any idea how to check if the probelm is NP hard or not in wireless comunication?

[Torsten]

any idea how to check if the probelm is NP hard or not in wireless comunication?

Which problem ?

[Alexa]

@Torsten

Max (1-exp(- SNR_threshold/average SNR))

subject to power constraints,0>p< P_max.

[Torsten]

I've never heard about an NP classification for this kind of problem.

[Alexa]

@Torsten Do you know how can check if it the mentioned probelm is convex or not?

[Torsten]

I don't understand the variables you try to optimize and I don't understand your constraints.

If you wrote

max(1-exp(-x))

under the constraint

0 <= x <= P_max

I'd immediately say: x = P_max is the solution.

But I can't tell what is hidden behind your problem formulation

Max (1-exp(- SNR_threshold/average SNR)) 

subject to power constraints,

0>p< P_max

.

[Alexa]

@Torsten

Okay, thank you. I am trying to maximize the probability that the SNR is greater than the SNR threshold, and my variable is the power allocation(p) and the P_max is the totoal power. This is ((1-exp(- SNR_threshold/average SNR)) ) to find the probability of each SNR.