FindMaximum[{1000 r1 + 1/2 r2 + 1/3 r3,
  {1000^2 r1 + 1/4 r2 + 1/9 r3 == 1/9,
   r1 + r2 + r3 + r4 == 2,
   r1^2 <= p1, r2^2 <= p2, r3^2 <= p3, r4^2 <= p4,
   1000^2 p1 + 1/4 p2 + 1/9 p3 == 1/9,
   p1 + p2 + p3 + p4 == 2,
   0 <= p1 <= 1, 0 <= p2 <= 1, 0 <= p3 <= 1, 0 <= p4 <= 1}
  },
  {{r1, 0}, {r2, 0}, {r3, 1}, {r4, 1}, {p1, 0}, {p2, 0}, {p3, 1}, {p4, 1}}, AccuracyGoal -> 10]

{0.45521, {r1 -> -6.30073*10^-8, r2 -> 0.268328, r3 -> 0.963328,
  r4 -> 0.768345, p1 -> 0., p2 -> 0.0719999, p3 -> 0.928, p4 -> 1.}}