What's the name of the problem of finding equilibrium concentrations? Is there any specific name of the problem of finding equilibrium concentrations with known equilibrium constants and initial concentrations? 

Would it be appropriate to call it "direct equilibrium problem", by analogy with the similar problem in chemical kinetics--finding concentrations as function of time for known rate constants--that is called "direct kinetic problem"?
I don't think the phrase "direct equilibrium problem" is used in this context.

"Multiple-equilibrium problem" would be appropriate.

As in the phrase "It involves solution of the multiple equilibrium problem" J. Molecular Biology vol. 247 page 774.

Or "chemical equilibrium problem" if you don't want to imply that there is more than one equilibrium.