Hello, I am now struggling to express the formula below to the legal operation form in CVX.
1/log2(1+a*exp(-norm(\mathbf{q}-\mathbf{u})) / norm(\mathbf{q}-\mathbf{u})^2) where \mathbf{q} and \mathbf{u} are 3D Cartsian coordinate and \mathbf{q}=\{x,y,H\} is optimization variable. Others are given (a, \mathbf{u}, H).
Note that I prove that the function is convex only when 0<(1+a*exp(-norm(\mathbf{q}-\mathbf{u})) / norm(\mathbf{q}-\mathbf{u})^2)<(some positive value). I need the help.
In addition, can it be solved via CVX directly by adding constraint 0<(1+a*exp(-norm(\mathbf{q}-\mathbf{u})) / norm(\mathbf{q}-\mathbf{u})^2)<(some positive value)?
It cannot be solved by adding the last constraint explicitly.
Then, this problem cannot utilize CVX for solving?
Probably yes, or no. Can you show your proof of convexity?
The details are difficult to show here, just check positivity of principal minors.
could you explain why it can or not?
Even if it is convex, that does not mean it can necessarily be formulated in CVX.
That’s why I request. Is there any chance to formulate this problem for CVX in an appropriate way? please let me know. thank you.
It doesn’t appear likely that any forum readers are going to come up with a CVX formulation for your problem. nevertheless,. a free virtual beer to anyone who does.
Assume x > 0, then \frac{\partial ^2\frac{1}{\log \left(\frac{\exp (-x)}{x^2}+1\right)}}{\partial x^2}=\frac{2 (x+2)^2-\left(e^x \left(x^2+4 x+6\right) x^2+2\right) \log \left(\frac{e^{-x}}{x^2}+1\right)}{\left(e^x x^3+x\right)^2 \log ^3\left(\frac{e^{-x}}{x^2}+1\right)}, but \underset{x\to 0.1}{\text{lim}}\left(2 (x+2)^2-\left(e^x \left(x^2+4 x+6\right) x^2+2\right) \log \left(\frac{e^{-x}}{x^2}+1\right)\right)=-0.532254<0, \underset{x\to 1}{\text{lim}}\left(2 (x+2)^2-\left(e^x \left(x^2+4 x+6\right) x^2+2\right) \log \left(\frac{e^{-x}}{x^2}+1\right)\right)=8.00661>0. So the function provided is neither convex nor concave.
