Find Max Instance Over Permutations

Question

I would like to find the maximum of some objective function over all possible permutations

(* Memory inefficient solution *)
Max[obj/@Permutations[lst]]

(* Outout: SystemException["MemoryAllocationFailure"] *)

While this computation might be intractably slow $O(n!)$, it should have $O(n)$ space for lst of size $n$.

Is there an efficient way to only cache the max seen so far? Thanks

Here is a standalone trivial example for the purposes of testing

n = 10;
l = Range[n];

Max[l . # & /@ Permutations[l]]
(* 385 *)
```