I try to solve this 0-1 integer linear program (maximize, thus the minus) with LinearOptimization:

LinearOptimization[
 -(15 x + 6 y + 0 a),
 {
  275 a >= 0,
  a <= x,
  a <= y,
  a >= -1 + x + y,
  
  a >= 0,
  a <= 1,
  
  x >= 0,
  x <= 1,
  
  y >= 0,
  y <= 1
  }, {x, y, a}, Integers
 ]

This should yield the solution x = y = a = 1. However, I get the following error:

LinearOptimization::misupp: The CLP method does not support mixed integer optimization.

I introduced a to linearize the constraint 275 x y >= 0. Also, I used matrix/vector notation in LinearOptimization

LinearOptimization[-{15, 6, 0}, {
  -{{0, 0, -275},
    {-1, 0, 1},
    {0, -1, 1},
    {1, 1, -1},
    {-1, 0, 0},
    {1, 0, 0},
    {0, -1, 0},
    {0, 1, 0},
    {0, 0, -1},
    {0, 0, 1}},
  {0, 0, 0, 1, 0, 1, 0, 1, 0, 1}}]

This works and returns {1,1,1}.

The bottom approach shows that my constraints are linear, so why does Mathematica think my program is mixed-linear?