Is there any way to tell Solve (or something similar) to return values for variables that solved as many constraints as it happened to satisfy?

###Edit:

I had trouble finding a small example that wasn't trivial, but I finally did. Here's one:

Solve[{{-1 + Abs[q2 (-(7/4) + t1)], q1, -1 + Abs[q1 (-(7/4) + t2)], 3 + q2} == 0,
       {t1, t2} >= 7/4},
      {t1, t2, q1, q2}]

The kind of output that I want (I don't care if it's not unique; I can deal with that):

{{t1 -> 25/12, t2 -> Indeterminate, q1 -> 0, q2 -> -3}}

The output that I get, but don't want:

{}

Is there any way to tell Solve (or something similar) to return values for variables that solved as many constraints as it happened to satisfy?

Edit:

I had trouble finding a small example that wasn't trivial, but I finally did. Here's one:

Solve[{{-1 + Abs[q2 (-(7/4) + t1)], q1, -1 + Abs[q1 (-(7/4) + t2)], 3 + q2} == 0,
       {t1, t2} >= 7/4},
      {t1, t2, q1, q2}]

The kind of output that I want (I don't care if it's not unique; I can deal with that):

{{t1 -> 25/12, t2 -> Indeterminate, q1 -> 0, q2 -> -3}}

The output that I get, but don't want:

{}

Is there any way to tell Solve (or something similar) to return values for variables that solved as many constraints as it happened to satisfy?

Right now I'm reformulating the problem as an optimization problem and using ArgMin to minimize the norm of the error vector, but I'm wondering if there is a faster/better way, since I imagine Solve is presumably already doing some of the required work regardless, and ArgMin is a bit overkill and too slow.

Is there any way to tell Solve (or something similar) to return values for variables that solved as many constraints as it happened to satisfy?

###Edit:

I had trouble finding a small example that wasn't trivial, but I finally did. Here's one:

Solve[{{-1 + Abs[q2 (-(7/4) + t1)], q1, -1 + Abs[q1 (-(7/4) + t2)], 3 + q2} == 0,
       {t1, t2} >= 7/4},
      {t1, t2, q1, q2}]

The kind of output that I want (I don't care if it's not unique; I can deal with that):

{{t1 -> 25/12, t2 -> Indeterminate, q1 -> 0, q2 -> -3}}

The output that I get, but don't want:

{}