FindMaximum::notlm
The first argument has an incorrect form for use with Method->"LevenbergMarquardt":
FindMaximum[x ^ 2 - 4x - (y - 3) ^ 2, {x, 1}, {y, 1}, Method -> "LevenbergMarquardt"]This shows a valid objective function for use with Method->LevenbergMarquardt:
FindMaximum[4 - (x - 2) ^ 2 - (y - 3) ^ 2, {x, 1}, {y, 1}, Method -> "LevenbergMarquardt"]