On the Optimal Integer-Forcing Precoding: A Geometric Perspective and a Polynomial-Time Algorithm

The joint optimization of the integer matrix $\mathbf{A}$ and the power scaling matrix $\mathbf{D}$ is central to achieving the capacity-approaching performance of Integer-Forcing (IF) precoding. This problem, however, is known to be NP-hard, presenting a fundamental computational bottleneck. In this paper, we reveal that the solution space of this problem admits a intrinsic geometric structure: it can be partitioned into a finite number of conical regions, each associated with a distinct full-rank integer matrix $\mathbf{A}$. Leveraging this decomposition, we transform the NP-hard problem into a search over these regions and propose the Multi-Cone Nested Stochastic Pattern Search (MCN-SPS) algorithm. Our main theoretical result is that MCN-SPS finds a near-optimal solution with a computational complexity of $\mathcal{O}\left(K^4\log K\log_2(r_0)\right)$, which is polynomial in the number of users $K$. Numerical simulations corroborate the theoretical analysis and demonstrate the algorithm's efficacy.