Reliable Projection Based Unsupervised Learning for Semi-Definite QCQP with Application of Beamforming Optimization

TL;DR

This work tackles a NP-hard QCQP for downlink beamforming with QoS constraints, formalized as $\min_{w} \|w\|^2$ subject to $\frac{\|h_i^H w\|^2}{\sigma_i^2} \ge \gamma_i$. It introduces a reliable, differentiable projection that maps neural network outputs into the feasible region by scaling along the origin with $t = \max_i \sqrt{\frac{\gamma_i \sigma_i^2}{\|h_i^H w\|^2}}$, enabling a boundary-focused solution via $x = t w$, which guarantees feasibility for all constraints. The projection is differentiable, supporting a label-free unsupervised training approach by deriving gradients through $t$ and the projection, reducing data-label requirements. By transforming to an unconstrained formulation, the method reduces neural network complexity and accelerates convergence, with an MLP of input size $2NM$ and output size $2N$ achieving $O(NM)$ inference and $O(M)$ projection costs. Empirical results show the projection-based L2O approach attains performance close to the SDR lower bound while offering faster inference and better scalability, demonstrating practical viability for real-time beamforming in 6G networks.

Abstract

In this paper, we investigate a special class of quadratic-constrained quadratic programming (QCQP) with semi-definite constraints. Traditionally, since such a problem is non-convex and N-hard, the neural network (NN) is regarded as a promising method to obtain a high-performing solution. However, due to the inherent prediction error, it is challenging to ensure all solution output by the NN is feasible. Although some existing methods propose some naive methods, they only focus on reducing the constraint violation probability, where not all solutions are feasibly guaranteed. To deal with the above challenge, in this paper a computing efficient and reliable projection is proposed, where all solution output by the NN are ensured to be feasible. Moreover, unsupervised learning is used, so the NN can be trained effectively and efficiently without labels. Theoretically, the solution of the NN after projection is proven to be feasible, and we also prove the projection method can enhance the convergence performance and speed of the NN. To evaluate our proposed method, the quality of service (QoS)-contained beamforming scenario is studied, where the simulation results show the proposed method can achieve high-performance which is competitive with the lower bound.

Figures (3)

• Figure 1: Inference and training procedure.
• Figure 2: Illustration of projection.
• Figure 9: Caption for the whole figure.

Theorems & Definitions (14)

• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 1
• proof
• Corollary 1
• proof
• Corollary 2
• proof