  We consider the problem of max-min fairness for uplink cell-free massive
multiple-input multiple-output (MIMO) subject to per-user power constraints.
The standard framework for solving the considered problem is to separately
solve two subproblems: the receiver filter coefficient design and the power
control problem. While the former has a closed-form solution, the latter has
been solved using either second-order methods of high computational complexity
or a first-order method that provides an approximate solution. To deal with
these drawbacks of the existing methods, we propose a mirror prox based method
for the power control problem by equivalently reformulating it as a
convex-concave problem and applying the mirror prox algorithm to find a saddle
point. The simulation results establish the optimality of the proposed solution
and demonstrate that it is more efficient than the known methods. We also
conclude that for large-scale cell-free massive MIMO, joint optimization of
linear receive combining and power control provides significantly better user
fairness than the power control only scheme in which receiver coefficients are
fixed to unity.
