  This paper investigates a general discrete $\ell_p$-norm maximization
problem, with the power enhancement at steering directions through
reconfigurable intelligent surfaces (RISs) as an instance. We propose a
mathematically concise iterative framework composed of alternating inner
product maximizations, well-suited for addressing $\ell_1$- and $\ell_2$-norm
maximizations with either discrete or continuous uni-modular variable
constraints. The iteration is proven to be monotonically non-decreasing.
Moreover, this framework exhibits a distinctive capability to mitigate
performance degradation due to discrete quantization, establishing it as the
first post-rounding lifting approach applicable to any algorithm intended for
the continuous solution. Additionally, as an integral component of the
alternating iterations framework, we present a divide-and-sort (DaS) method to
tackle the discrete inner product maximization problem. In the realm of
$\ell_\infty$-norm maximization with discrete uni-modular constraints, the DaS
ensures the identification of the global optimum with polynomial search
complexity. We validate the effectiveness of the alternating inner product
maximization framework in beamforming through RISs using both numerical
experiments and field trials on prototypes. The results demonstrate that the
proposed approach achieves higher power enhancement and outperforms other
competitors. Finally, we show that discrete phase configurations with moderate
quantization bits (e.g., 4-bit) exhibit comparable performance to continuous
configurations in terms of power gains.
