I am relatively new to convex optimization and am looking to solve a resource allocation problem. I understand, that if my utility function is concave the following problem constitutes "an optimization of a concave function over a convex set" and should therefore have a unique solution (as an example I am looking at proportional fairness, for which utility is the logarithmic function):
\begin{array}{rrclcl} \displaystyle \max_{x} & {\sum_{c \in C} U_c(x_c)} \\ \textrm{s.t.} & \mathbf{A} \mathbf{x} & \leq & \mathbf{r} \\ & x_c & \geq & 0 & & \forall c \in C \\\\ \end{array}
Now my problem is that in my system I have users who will get a linear combination of these resources assigned. Therefore their utility is based on the sum of resources they are assigned.
I have tried to incorporate this into the problem definition, but am not sure if this is a correct way to specify an optimization problem:
\begin{array}{rrclcl} \displaystyle \max_{y} & {\sum_{c \in C} U_c(y_c)} \\ \textrm{s.t.} & \mathbf{A} \mathbf{x} & \leq & \mathbf{r} \\ & \mathbf{B} \mathbf{x} & = & \mathbf{y} \\ & x_c & \geq & 0 & & \forall c \in C \\\\ \end{array}
I believe that the statement is mathematically correct, but would like to ask:
Is this formulation "standard compliant"? Should it be formulated differently to be solvable e.g. by standard optimization tools?
If yes, (how) can this be done?
Does this problem allow for statements about solvability/ number of solutions?
I am also happy for any pointers to related problems or terminology describing my problem - I can currently not find anything online although I am convinced that this is not an unusual problem.