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In this paper, we propose a new three-user network information flow model, referred to as the triangular multiple-input–multiple-output (MIMO) relay (TMR) channel, which consists of three users and three relays equipped with nU and nR antennas, respectively. Each user sends two independent messages to the other two users via the adjacent relays in two time slots, which are referred to as the multiple-access and broadcast stages. We derive a novel simultaneous signal and interference alignment for the proposed TMR channel in a scenario where there are fewer antennas at each relay than at each user (nR < nU ) . An optimized pseudo-inverse scheme based on an efficient gradient projection algorithm is proposed to solve the simultaneous alignment problem. By deriving a gradient over weighted sum-rate maximization and applying a gradient descent method, the optimal beamforming vectors are obtained to maximize the effective signal-to-noise ratios. Furthermore, to obtain rapid convergence speed and reduce computational complexity, we introduce a quasi-Newton method, which is referred to as the Broyden–Fletcher–Goldfarb–Shanno algorithm, by approximating the Hessian matrix of a pure Newton method. The convergence of the proposed gradient algorithm is guaranteed by proposing a line search algorithm. Finally, a performance evaluation shows that the proposed scheme offers a higher sum rate, produces a better outage probability, and achieves a higher multiplexing gain than the existing schemes. [ABSTRACT FROM PUBLISHER]