How to Minimize a problem with a big number of variables?

Using a different approach: guess the sequence. Then the answer for 2019 returns the right result in 20 microseconds. However, it does not prove that the sequence is the actual answer.

sol = ParallelTable[{n, Minimize[{Sum[x[j], {j, 1, n}], 
      Table[x[j] + x[j + 1] >= j, {j, 1, n - 1}], x[n] + x[1] >= n}, 
     Table[x[j], {j, 1, n}]]}, {n, 2, 50}];

f = FindSequenceFunction[sol[[All, 2, 1]]];
answer[n_] = f[n - 1];
(* 1/8 (-1)^(-1 + n) (-1 + 5 (-1)^(-1 + n) + 2 (-1)^(-1 + n) (-1 + n)) n *)

answer[2019]
(* 1019595 *)

NMinimize calculates the case n==2019

 mini[n_] :=NMinimize[{Sum[x[j], {j, 1, n}],Table[x[j] + x[j + 1] >= j, {j, 1, n - 1}], x[n] + x[1] >= n},Table[x[j], {j, 1, n}]][[1]]
 
 mini[2019] // Rationalize // AbsoluteTiming
 (*{0.118309, 1019595}*)

in .11 seconds!

n = 2019;
c = Table[1, n];
m = SparseArray[{Band[{1, 1}] -> 1, Band[{1, 2}] -> 1, {n, 1} -> 1}, {n, n}];
b = Range[n];
LinearOptimization[N[c], {m, -b}, "PrimalMinimumValue"] // AbsoluteTiming // DecimalForm
(* {0.0275781, 1019595.} *)

Note that the use of N to force machine precision is important, otherwise the result will be exact but much slower. For some reason, the slowdown becomes much more significant at n = 201:

           Exact   Approximate
n = 200    0.35 s     0.002 s
n = 201    8.9 s      0.002 s

---

Using LinearProgramming gives the same results (and has the same timing issue):

Total @ LinearProgramming[N[c], m, b, -∞] // AbsoluteTiming // DecimalForm
(* {0.0286999, 1019595.} *)

Note the -∞, without it the constraint x ≥ 0 is added.