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One of the downsides of the massive multiple-input-multiple-output (M-MIMO) system is its computational complexity. Considering that techniques and different algorithms proposed in the literature applied to conventional MIMO may not be well suited or readily applicable to M-MIMO systems, in this paper, the application of different formulations inside the convex optimization framework is investigated. This paper is divided into two parts. In the first part, linear programming, quadratic programming (QP), and semidefinite programming are explored in an M-MIMO environment with high-order modulation and under realistic channel conditions, i.e., considering spatial correlation, error in the channel estimation, as well as different system loading. The bit error rate is evaluated numerically through Monte Carlo simulations. In the second part, algorithms to solve the QP formulation are explored, and computational complexity in terms of floating-point operations (flops) is compared with linear detectors. Those algorithms have interesting aspects when applied to our specific problem (M-MIMO detection formulated as QP), such as the exploitation of the structure of the problem ( simple constraints ) and the improvement of the rate of convergence due to the well-conditioned Gram matrix (channel hardening). The number of iterations is higher when the number of users    $K$    becomes similar to the number of base station antennas    $M$    (i.e.,    $Kapprox M$   ) than the case    $Kll M$   ; the number of iterations increases slowly as the number of users    $K$    and base station antennas    $M$    increases while keeping a low system loading. The QP with projected algorithms presented better performance than minimum mean square error detector when    $Kapprox M$    and promising computational complexity for scenarios with increasing    $K$    and low system loading.

Linear, Quadratic, and Semidefinite Programming Massive MIMO Detectors: Reliability and Complexity