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.

翻译：整数矩阵 $\mathbf{A}$ 与功率缩放矩阵 $\mathbf{D}$ 的联合优化是实现整数迫零预编码逼近容量性能的关键。然而，该问题已知是 NP 难问题，构成了一个根本性的计算瓶颈。本文揭示了该问题的解空间具有内在的几何结构：它可以被划分为有限个锥形区域，每个区域与一个特定的满秩整数矩阵 $\mathbf{A}$ 相关联。利用这种分解，我们将 NP 难问题转化为对这些区域的搜索，并提出了多锥嵌套随机模式搜索算法。我们的主要理论结果表明，MCN-SPS 能以 $\mathcal{O}\left(K^4\log K\log_2(r_0)\right)$ 的计算复杂度找到一个近似最优解，该复杂度在用户数 $K$ 上是多项式的。数值仿真验证了理论分析，并证明了该算法的有效性。