Unable to generate any initial points with NMaximize (or NMinimize)

[Meike]

I often get the error:

NMaximize::incst: NMaximize was unable to generate any initial points satisfying the inequality constraints

while there are clearly points satisfying the inequality constraints. For example:

NMaximize[{1/(0.` + (0.012710615354858255` (0.07879657698200958` +
a) (0.000046296899009293725` +
0.003958158898356454` b) (233.02777267494722` +
102.02996575509754` fm))/(a (4.892449413727213`*^-6 -
9.784898827454427`*^-7 fm - 0.4123000838756304` b fm)) + (
0.10155110608580004` (0.0367973376747971` +
c) (0.5030193912642286` + 16.257272679476753` b +
0.0309411917473244` p))/b + (
219.22680202984347` (0.03970570642848887` +
0.013422101464134875` fm) (0.07972967880969935` +
0.14421246041749264` p))/(fm p)), a > $MachineEpsilon,
b > $MachineEpsilon, fm > $MachineEpsilon, c > $MachineEpsilon,
p > $MachineEpsilon,
0.00009600189408524172` (5 - fm) -
40.451710009005595` b fm > $MachineEpsilon, fm < 5.`}, {a, b, fm,
c, p}, MaxIterations -> 1000]

NMaximize::incst: NMaximize was unable to generate any initial points satisfying the inequality constraints {-0.000480009+0.0000960019 fm+40.4517 b fm<=0}. The initial region specified may not contain any feasible points. Changing the initial region or specifying explicit initial points may provide a better solution. >>

Clearly there are initial points, for example:

0.00009600189408524172` (5 - fm) -
40.451710009005595` b fm > $MachineEpsilon /. {b -> 0.0007,
fm -> 0.00003}

True

Also, if I make a more simple example with the same constraint I do not get the same error. I know I could give initial values myself, but since I'm automatically generating many of the optimizations, I would prefer not to. And I would like to understand why I keep getting this error and if there is any way to solve it.