I am working on optimization of a function of (variable h) i.e,

max f(h)=log ( h^2 + a^2 ) * ( a + 2 sqrt (h^2+a^2) ) ,

where a and h both are non negative.

Also ‘(h^2 + a^2)’ is convex inside the log function and ‘sqrt (h^2+a^2)’ is also convex.

I also proved that the second derivative of this function ‘f(h)’ itself is positive. Therefore I wanted to write it in CVX but it is giving me an error 'invalid log(convex ) ’ function. Please guide me if there is a way to write it in cvx.

My calculations in MAPLE show that this is not convex for all nonnegative a and h. For instance, the 2nd derivative w.r.t. h is negative for a= 0.1, h = 0.01.

x^2 + a^2 rears it ugly head again. If this can not be handled by one of the Mosek posters, it’s another candidate for @Erling’s convex modeling challenge Express 1/(a^2+x^2) in cvx

Note that in this case, with a, which is an input value (i.e., constant) suitably restricted, the function appears to be convex over the entirety of its domain.

BTW, your plot is not accepted on this forum as being a proof. I think it is convex for a >= 0.23, but I haven’t proven it.

I tried to write the problem by addition of some other terms now the problem looks like this

max f(h)=z/y;
where
z=- log ( h^2 + a^2 ) * ( M * (a + 2 sqrt (h^2+a^2))-N h ) ,

, y=P * (a + 2 sqrt (h^2+a^2))-Q h
, N=8.31,M=5.19,P=1.76;Q=2.28. I proved that for a>2 the function z is concave and y is alwasys convex. There are references that says global optimal exists but I still could not write it in CVX . Please guide me thanks.