Lagrangian-Constrained deep neural network for joint relay selection and power allocation in two-hop AF cooperative networks

Power and size constrained nodes in fifth generation (5G) and beyond-5G wireless networks preclude the direct deployment of multiple-input multiple-output (MIMO) transceivers for spatial diversity. Cooperative communication offers an effective alternative by exploiting the broadcast nature of the wireless channel, enabling geographically distributed single antenna nodes to form a virtual antenna array. This paper proposes a Lagrangian-constrained deep neural network (LC-DNN) framework for joint relay selection and power allocation in a two-hop multi-relay Amplify-and-Forward (AF) cooperative network. The LC-DNN model accepts normalised composite channel gains, incorporating channel state information (CSI) and inter-node path loss as its input feature vector and directly outputs the source transmission power and per-relay forwarding coefficients that maximise the end-to-end data rate subject to explicit power and rate constraints embedded as Lagrangian penalty terms in the loss function. A two-hidden-layer fully connected architecture (256–256 neurons, ReLU activations, sigmoid activation output) trained on 100,000 independent Rayleigh fading channel realisations achieved a test accuracy of 83.66% and final training loss of 0.0218 surpasses deeper configurations which exhibit overfitting. An optimized two-hidden-layer fully connected network is trained using independently generated Rayleigh-fading channel realizations, and its performance is evaluated under individual relay-power constraints of 50, 75, and 100 mW with an aggregate budget of 1000 mW. Under a network of 20 relays with a 100 mW per-relay power constraint, LC-DNN achieves a test accuracy of 83.5% ± 0.7%, with the per-relay power constraint satisfied in 100% of test samples. LC-DNN achieves 7.82 dB SNR gain over a prior DNN-based baseline [Guo et al., 2022] and a 0.92 dB gain over equal-power allocation, measured at the outage probability 10⁻³. Network parameter count grows only 16% as the relay count increases fourfold (M = 5 to 20), with measured inference time under 0.12 ms across this range, confirming that inference cost is dominated by the fixed hidden layer width rather than the number of relays.