The Interference-Conjugate Transform: The Tightest Interference-Linear Surrogate for Power Control, Scheduling and Beamforming in Wireless Networks
Abstract: Closed-form fractional programming (FP), weighted MMSE (WMMSE), and the 2026 surrogate-enhanced methods SEFP and XMMSE all maximize the weighted sum rate by iteratively maximizing a surrogate linear in the interference power. We show that this family admits a unique best member: the interference-conjugate transform (ICT), which replaces each rate term $\log(1+a/b)$ by $\max_{κ\ge0}[Ω(κa)-κb]$, where $Ω$ is the closed-form partial convex conjugate of the rate in the interference variable and the optimal multiplier is the interference price $a/(b(a+b))$. We prove that, for any fixed price, $Ω(κ\,\cdot)$ is the pointwise largest own-signal function compatible with a valid interference-linear lower bound, and that tightness at an anchor forces the price. Hence all four benchmark surrogates, which share this price and differ only in their own-signal term, are dominated pointwise by ICT. At SINR $γ$, the benchmarks' own-signal curvature exceeds the true curvature by factors behaving as $γ/2$ (FP/WMMSE) and $(\logγ)/2$ (SEFP/XMMSE), against $1+1/(γ(γ+2))$ for ICT, and this ordering carries over to the certified ascent per iteration and the asymptotic linear convergence rate. ICT yields closed-form updates at the benchmarks' per-iteration cost for single-antenna power control, uplink joint scheduling and power control, and multi-cell MISO beamforming, and the first solver-free algorithm for sum-least-$q$th-percentile rate maximization, via a $K$-dimensional convex dual with closed-form gradient and Hessian and a certified duality gap. In uplink scheduling, ICT improves proportional-fairness utility over FP by +7.03 and over SEFP by +3.35 (60/60 drops) using 18.7 iterations per slot vs. 49.7 and 38.0; in MISO beamforming with $M=8$, it reaches 99.9% of its final value in a median of 9 iterations vs. 16 (XMMSE) and 1160 (FP/WMMSE).