I am trying to find the maximum of a function $f(x,y)$ in which $x$ and $y$ are dependent on each other. For example, $x$ is the size and $y$ is the weight of a component.
In order to find the maximum value of this unconstrained optimization problem, first I need to make sure that the function $f$ is jointly concave with respect to $(x,y)$. When I compute the Hessian matrix of $f(x,y)$, my function is not jointly concave because decision variables are dependent.
Can anyone provide me a recommendation on how can I find the optimal value of this function?
One way is to solve for one in terms of the other from your constraints. In other words, if $y=y(x)$ then $$ f(x,y) = f(x,y(x)) $$ which is now an ordinary function of one variable.
Alternatively, you can look for optima even in non-jointly-concave functions, just it is a little trickier to find global optima that way...