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We consider the problem of max-min fairness for uplink cell-free massive multiple-input multiple-output which is a potential technology for beyond 5G networks. More specifically, we aim to maximize the minimum spectral efficiency of all users subject to the per-user power constraint, assuming linear receive combining technique at access points. The considered problem can be further divided into two subproblems: the receiver filter coefficient design and the power control problem. While the receiver coefficient design turns out to be a generalized eigenvalue problem, and thus, admits a closed-form solution, the power control problem is numerically troublesome. To solve the power control problem, existing approaches rely on geometric programming (GP) which is not suitable for large-scale systems. To overcome the high-complexity issue of the GP method, we first reformulate the power control problem intro a convex program, and then apply a smoothing technique in combination with an accelerated projected gradient method to solve it. The simulation results demonstrate that the proposed solution can achieve almost the same objective but in much lesser time than the existing GP-based method.