Existing phase optimization methods in reconfigurable intelligent surfaces (RISs) face significant challenges in achieving flexible beam synthesis, especially for directional beam suppression. This paper introduces a Max-min criterion incorporating non-linear constraints, utilizing optimization techniques to enable multi-beam enhancement and suppression via transmissive RISs. A realistic model grounded in geometrical optics is first presented to characterize the input/output behavior of transmissive RIS, effectively linking explicit beam-forming operations with practical implementation. Subsequently, a highly efficient bisection-based algorithm for constrained Max-min optimization involving quadratic forms is developed, utilizing an auxiliary variable and Moreau envelope to iteratively reach the optimal solution. This approach demonstrates excellent extensibility and is applicable to a wide range of constrained Max-min problems. Numerical simulations validate the proposed methods, confirming that the framework enables beam enhancement or suppression at designated spatial positions.

翻译：现有可重构智能表面（RIS）的相位优化方法在实现灵活波束合成方面面临重大挑战，特别是在定向波束抑制方面。本文提出了一种结合非线性约束的Max-min准则，利用优化技术通过透射式RIS实现多波束增强与抑制。首先，基于几何光学建立了一个实际模型来表征透射式RIS的输入/输出行为，从而有效将显式波束成形操作与实际实现联系起来。随后，开发了一种高效的基于二分法的约束Max-min优化算法，该算法涉及二次型，通过引入辅助变量和Moreau包络迭代逼近最优解。该方法展现出优异的可扩展性，适用于广泛的约束Max-min问题。数值仿真验证了所提方法的有效性，证实该框架能够在指定空间位置实现波束增强或抑制。