Duality between the Power Minimization and Max-Min SINR Balancing Symbol-Level Precoding

This paper reveals the latent relation inherent in two typical problems in constructive interference (CI)-based symbol-level precoding (SLP). One is the power minimization (PM) problem subject to instantaneous signal-to-interference-plus-noise ratio (SINR) constraints, and the other is the weighted max-min SINR balancing (SB) problem with the symbol-level transmit power budget. In particular, we establish an explicit duality between the PM-SLP and SB-SLP problems, where we prove that one of the two problems can be uniquely mapped to the other. The proposed duality not only provides insights into the intrinsic structure of the problems and solutions but also facilitates obtaining the solution to the SB-SLP given the solution to the PM-SLP without the need for one-dimension search, and vice versa. We further propose a closed-form power scaling algorithm to solve the SB-SLP via PM-SLP, by which the separability of the PM-SLP can be leveraged to solve the two problems simultaneously. Numerical results demonstrate our derivations on the duality as well as the efficiency of the proposed power scaling algorithm.