I want to find the maximum of following \begin{align} f(X,Y)=\max_{x_k, y_k} \sum_{k=1}^L log(1+{x_k}\frac{py_k}{1+py_k}) \\s.t. \sum_{k=1}^L x_k=La, \\ \sum_{k=1}^L y_k=Lb \\ a>0, ~\ b>0, ~\ p>0, \\ x_1\ge x_2\ge ... \ge x_L \ge 0\text{ and } y_1\ge y_2\ge ... \ge y_L \ge 0 \end{align}
I have some intuition it might be upper bounded by \begin{align} Llog(1+\frac{pab}{1+pb}) \end{align} where \begin{align} x_1=x_2=...x_L=a ~\ and ~\ y_1=y_2=...=y_L=b \end{align}
is it right? if so how can I prove this rigorously? or how can I find upper bound and the condition satisfying maximum??
thanks!
Now we can.
Hence, the objective function:
$$ f=\sum_{i=1}^N log (1+x_i\frac{py_i}{1+py_i}) $$
With the constraints:
$$ g^1=\sum_{i=1}^N x_i-La\\ g^2=\sum_{i=1}^N y_i-Lb $$
Hence the 2L dimension gradients for these functions both for $x_i$ and $y_i$ are (check it!!):
$$ \nabla f_{x_i}=\frac{1}{1+x_i\frac{py_i}{1+py_i}}\frac{py_i}{1+py_i}\\ \nabla f_{y_i}=\frac{1}{1+x_i\frac{py_i}{1+py_i}}\frac{x_i}{(1+py_i)^2}\\ \nabla g^1_{x_i}=1\\ \nabla g^1_{y_i}=0\\ \nabla g^2_{x_i}=0\\ \nabla g^2_{y_i}=1 $$
Hence the optimal condition is the void sum of all that 2L vectors:
$$ -\nabla f_{x_i}+\lambda^1\nabla g^1_{x_i}+\lambda^2\nabla g^2_{x_i}=0\\ -\nabla f_{y_i}+\lambda^1\nabla g^1_{y_i}+\lambda^2\nabla g^2_{y_i}=0 $$
Which is:
$$ \frac{1}{1+x_i\frac{py_i}{1+py_i}}\frac{py_i}{1+py_i}=\lambda^1\\ \frac{1}{1+x_i\frac{py_i}{1+py_i}}\frac{x_i}{(1+py_i)^2}=\lambda^2\\ $$
Hence we have 2L gradient equations, and 2 constraints equations, for 2L coordinates and 2 lagrange multipliers.
If we assume any $\lambda$ is zero, any of the constraints will not hold. Thus both multipliers are nonzero, hence both constraints are active.
As a helper, by dividing both gradient equations, we isolate the $x_i$:
$$ x_i=\frac{\lambda^2}{\lambda^1}py_i(1+py_i) $$
And back in the gradient now we isolate $y_i$:
$$ \frac{1}{\lambda^1+\lambda^2py_i^2}\frac{py_i}{1+py_i}=1 $$
Both of these sets of equations, together eith the constraints, at this point, un-isolatable:
$$ \sum_{i=1}^N x_i=La\\ \sum_{i=1}^N y_i=Lb $$
If everything was right, these expressions are for a 2-D simplex in a 2L-D space, following the non linear ecuations we have just obtained.
Because of the constraints, as far as this points indicates,a closed solution is not possible, leaving this problem as is for a CAS (solver), depending on the actual values of the constants