Let $S_n$ be the simplex in $n$-dimensions, i.e., the set $\{ (x_1, \ldots, x_n) ~|~ \sum_{i=1}^n x_i = 1, x_i \geq 0 \mbox{ for all } i \}$. I am interested in the optimization problem $ {\rm max} \sum_{i=1}^n c_i x_i $ subject to the two constraints $ x \in S_{n}, ~~~~\sum_{i=1}^n d_i x_i = 0.$
Does there exist a closed form expression for the optimal vector and for the optimal value?
Note that if all $d_i$ are zero then the optimal value is $\max_{i=1, \ldots, n} c_i$, achieved by taking $x_i=1$ corresponding to the largest $c_i$, and $x_i=0$ for other $i$. If the $d_i$ are all equal, then the set of feasible solutions is empty.