expr2 = (31 (52275 arb[1] + 17425 arb[2] -
   8659 (3 arb[3] + arb[4])))/(
181178952 (-arb[2] arb[3] + arb[1] arb[4])) >= 0 && (
31 (6531 arb[1] + 2177 arb[2] - 4283 (3 arb[3] + arb[4])))/(
181178952 (-arb[2] arb[3] + arb[1] arb[4])) >= 0 && (
31 (129 arb[1] + 43 arb[2] - 13 (3 arb[3] + arb[4])))/(
4213464 (-arb[2] arb[3] + arb[1] arb[4])) >= 0 && (
505325 arb[1] + 5035825 arb[2] - 8659 (29 arb[3] + 289 arb[4]))/(
60392984 (arb[2] arb[3] - arb[1] arb[4])) >= 0 && (
63133 arb[1] + 629153 arb[2] - 4283 (29 arb[3] + 289 arb[4]))/(
60392984 (arb[2] arb[3] - arb[1] arb[4])) >=
0 && (-1247 arb[1] - 12427 arb[2] + 377 arb[3] + 3757 arb[4])/(
1404488 (-arb[2] arb[3] + arb[1] arb[4])) >=
0 && (-557600 arb[1] + 2247825 arb[2] +
 8659 (32 arb[3] - 129 arb[4]))/(
90589476 (arb[2] arb[3] - arb[1] arb[4])) >=
0 && (-69664 arb[1] + 280833 arb[2] +
 4283 (32 arb[3] - 129 arb[4]))/(
90589476 (arb[2] arb[3] - arb[1] arb[4])) >= 0 && (
1376 arb[1] - 5547 arb[2] - 416 arb[3] + 1677 arb[4])/(
2106732 (-arb[2] arb[3] + arb[1] arb[4])) >=
0 && ((93 - 745 arb[2]) arb[3] + (31 + 745 arb[1]) arb[4])/(
838 (-arb[2] arb[3] + arb[1] arb[4])) >= 0 && (
31 (-arb[2] (-1 + arb[3]) + arb[1] (3 + arb[4])))/(
838 (arb[2] arb[3] - arb[1] arb[4])) >= 0 && (
31 (arb[2] (29 - 774 arb[3]) + 31 (3 arb[3] + arb[4]) +
   arb[1] (87 + 774 arb[4])))/(
216204 (-arb[2] arb[3] + arb[1] arb[4])) >= 0 && (
87 (1 + arb[2]) arb[3] + 3 (289 - 29 arb[1]) arb[4])/(
838 (arb[2] arb[3] - arb[1] arb[4])) >=
0 && (-867 arb[2] (-1 + arb[3]) + arb[1] (87 + 867 arb[4]))/(
838 (-arb[2] arb[3] + arb[1] arb[4])) >= 0 && (
arb[2] (8381 - 7482 arb[3]) + 899 arb[3] + 8959 arb[4] +
 29 arb[1] (29 + 258 arb[4]))/(
72068 (arb[2] arb[3] - arb[1] arb[4])) >=
0 && ((32 - 387 arb[2]) arb[3] + 129 (-1 + 3 arb[1]) arb[4])/(
419 (-arb[2] arb[3] + arb[1] arb[4])) >= 0 && (
129 arb[2] (-1 + arb[3]) + arb[1] (32 - 129 arb[4]))/(
419 (arb[2] arb[3] - arb[1] arb[4])) >= 0 && (
992 arb[3] - 129 arb[2] (29 + 64 arb[3]) - 3999 arb[4] +
 32 arb[1] (29 + 258 arb[4]))/(
108102 (-arb[2] arb[3] + arb[1] arb[4])) >= 0;

vars2 = arb /@ Range[4];

cons2 = And @@ Thread[-10 <= vars2 <= 10];

(soln3 = NMinimize[{Total[List @@ expr2[[All, 1]]], expr2 && cons2},
 vars2, Reals]) // Quiet // AbsoluteTiming

expr2 && cons2 /. soln3[[2]]

(soln4 = NMaximize[{Total[List @@ expr2[[All, 1]]], expr2 && cons2},
 vars2, Reals]) // Quiet // AbsoluteTiming

expr2 && cons2 /. soln4[[2]]

soleg1=Thread[vars2 -> {49/160, 46837/64500, 7071/4000, 25619/129000}];
soleg2=Thread[vars2 -> {151/2000, 88147/258000, 4019/4000, -(67417/516000)}];
expr2 && cons2 /. soleg1 (* Returns True *)
expr2 && cons2 /. soleg2 (* Returns True *)