It remains an open problem to find the optimal configuration of phase shifts under the discrete constraint for intelligent reflecting surface (IRS) in polynomial time. The above problem is widely believed to be difficult because it is not linked to any known combinatorial problems that can be solved efficiently. The branch-and-bound algorithms and the approximation algorithms constitute the best results in this area. Nevertheless, this work shows that the global optimum can actually be reached in linear time on average in terms of the number of reflective elements (REs) of IRS. The main idea is to geometrically interpret the discrete beamforming problem as choosing the optimal point on the unit circle. Although the number of possible combinations of phase shifts grows exponentially with the number of REs, it turns out that there are only a linear number of circular arcs that possibly contain the optimal point. Furthermore, the proposed algorithm can be viewed as a novel approach to a special case of the discrete quadratic program (QP).

翻译：在多项式时间内找到智能反射面（IRS）在离散约束下的最优相位配置仍是一个未解难题。上述问题普遍被认为难以解决，因为它与任何已知的可高效求解的组合优化问题均无关联。分支定界算法和近似算法构成了该领域的最佳研究成果。然而，本研究表明：以IRS反射单元（RE）数量为度量标准，全局最优解实际上可在平均线性时间内达成。核心思想是将离散波束成形问题几何化解释为在单位圆上选择最优点的过程。尽管相位组合的可能性数量随反射单元数量呈指数增长，但研究发现仅存在线性数量的圆弧可能包含最优点。此外，所提算法可视为离散二次规划（QP）特例的一种创新求解方案。