I am working on solving a multi-objective and I would appreciate some guidance and help.
I have basically three types of variables (inputs)
(i) Collections of variable of mixed variable type ==> $\overrightarrow{x} $
(ii) Multiple instantiation of single type ==> $\overrightarrow{\delta}, $ with component $ \delta_i$ where i = 1, 2, ....10
iii) Multiple instantiation of single type ==> $\overrightarrow{B}, $ with component B_j where j = 1, 2, .... 6
For any particular components of $B = B_1$, the output is given by:
$$\overrightarrow{y^1} = f(\overrightarrow{x}, \overrightarrow{\delta}, B_1) $$
Please notice that $\delta_i$ is directly mapped on ith component of $y_i^1$
Similar relation exists for other instantiation of $B = B_1, B_2, B_3 ... , B_6 $
$$\overrightarrow{y^2} = f(\overrightarrow{x}, \overrightarrow{\delta}, B_2) $$ $$\overrightarrow{y^3} = f(\overrightarrow{x}, \overrightarrow{\delta}, B_3) $$
.. $$\overrightarrow{y^6} = f(\overrightarrow{x}, \overrightarrow{\delta}, B_6) $$
We can stack these objective function column wise to make a multiobjective function.
Basically, the plot of data with $\overrightarrow{\delta} $ and B (dependence on $\overrightarrow{x}) not shown) looks like this:
It's reasonable to assume that all components are convex.
I am still in the planning stage of how to approach this optimization. So, I would appreciate some guidance from gurus. I would prefer some gradient based optimization. If the scalarization is suggested, then there is very good justification to allow for equal weighting of all terms.