Rethinking Fractional Programming for Joint Uplink Scheduling and Power Control in Multicell Wireless Networks

Abstract: This paper investigates the joint uplink scheduling and power control problem in a coordinated multicell wireless network, where at most one single-antenna user is allowed to access the single-antenna base station in each cell simultaneously. The resulting weighted sum-rate (WSR) maximization problem is a mixed discrete-continuous, nonconvex optimization problem that is notoriously difficult to solve directly. Classical fractional programming (FP) methods tackle this problem by leveraging the Lagrangian dual transform (LDT) followed by the quadratic transform (QT), yielding a tractable closed-form solution for scheduling and power control, with the LDT playing a crucial role in handling discrete variables. In this paper, we revisit the LDT from a minorization-maximization (MM) perspective and observe that its induced surrogate is somehow conservative due to the reciprocal-coordinate construction. Motivated by this observation, we propose a novel reciprocal-inversion transform (RIT) that constructs a tighter first-order Taylor expansion lower bound for the logarithmic rate function. The proposed RIT remains fully compatible with the QT, leading to a surrogate-enhanced FP (SEFP) algorithm for joint uplink scheduling and power control. The proposed SEFP algorithm retains the desirable per-cell separability of the classical FP framework and admits closed-form updates for the auxiliary variables, scheduling decisions, and transmit powers. Simulation results demonstrate that the SEFP algorithm consistently outperforms the classical FP method and other baselines for different network utilities.

• The paper introduces a novel RIT-based surrogate that tightens fractional programming for joint uplink scheduling and power control.
• It employs closed-form iterative updates achieving provable convergence and reduced computational complexity in multicell environments.
• Empirical results confirm significant performance gains over classical FP, especially under moderate-to-high SNR and diverse utility metrics.

Surrogate-Enhanced Fractional Programming for Joint Uplink Scheduling and Power Control

Introduction

This paper addresses the central challenge of joint user scheduling and power control in uplink multicell wireless networks. The optimization formulation is a nonconvex, mixed discrete-continuous weighted sum-rate (WSR) maximization problem, where each cell may schedule at most one single-antenna user for uplink access. The allocation problem is highly challenging due to the intricate coupling of intercell interference and the combinatorial nature of scheduling decisions. Historically, fractional programming (FP)—especially the Lagrangian Dual Transform (LDT) plus Quadratic Transform (QT) approach—has been pivotal in yielding tractable iterative algorithms with closed-form updates for both scheduling and power control, particularly when moving the signal-to-interference-plus-noise ratio (SINR) term outside the logarithm.

Problem Formulation and Classical FP Approach

The generic WSR objective is composed of a sum over $M$ terms, each involving a weighted log-rate:

$$F(\mathbf{x}) = \sum_{m=1}^{M} \omega_m \log \left( 1 + \frac{A_m(\mathbf{x})}{B_m(\mathbf{x})} \right),$$

with $\mathbf{x}$ denoting the vector of scheduling and power variables, and $(A_m, B_m)$ representing per-user signal and interference-plus-noise functions.

The closed-form FP technique applies the LDT to move the SINR outside the logarithm: $$F_\ell(\mathbf{x}, \boldsymbol{\gamma}) = \sum_{m=1}^M \omega_m \left[ \log(1 + \gamma_m) - \gamma_m + \frac{(1 + \gamma_m)A_m(\mathbf{x})}{A_m(\mathbf{x}) + B_m(\mathbf{x})}\right].$$ This is followed by the QT, decoupling fractional terms for efficient optimization across all users/cells. The per-iteration updates are provably convergent and offer computational efficiency in practice.

Limitations of LDT and Motivation for RIT

The paper provides a rigorous minorization-maximization (MM) interpretation of LDT, revealing structural conservativeness rooted in its reciprocal-coordinate construction. Specifically, the surrogate resulting from LDT:

• Lacks tightness at the $r=0$ boundary.
• Yields a surrogate curvature that is overly conservative (i.e., more negatively curved) compared to the original log-rate.
• Saturates for large $r$, failing to represent the unbounded growth of the log-rate at high SINR.

These limitations are a direct consequence of LDT’s variable substitution and indicate the necessity for a more flexible surrogate that preserves closed-form tractability but is less restrictive.

Reciprocal-Inversion Transform (RIT): Theoretical Framework

The paper introduces the Reciprocal-Inversion Transform (RIT), a new equivalent reformulation for the sum-of-logarithms objective. RIT leverages a different variable substitution ($\tau = 1/r$) and constructs a surrogate lower bound using first-order Taylor expansion in the reciprocal coordinate. The critical features are:

• The RIT-induced surrogate matches the value and first-order derivative of the original log-rate at the reference point.
• It provides a tighter approximation, especially at low and high SINR regions.
• The coefficients inside the fractional structure of the RIT surrogate are adaptive, depending on SINR, rather than fixed as in the LDT-based surrogate.

Combining RIT with the quadratic transform yields the Surrogate-Enhanced Fractional Programming (SEFP) algorithm. The resulting surrogate is provably tighter than that of LDT plus QT, as shown by explicit analytical comparison.

Application to Uplink Scheduling and Power Control

The SEFP is instantiated for the coordinated multicell uplink scenario with a single scheduled user per cell. For each iteration, the algorithm alternates updates of:

• RIT auxiliary variables (function of current user SINRs),
• QT auxiliary variables (closed-form for each user/cell),
• Scheduling and transmit power (per-cell maximization via a utility-minus-penalty metric with RIT-adaptive coefficients).

Each of these steps is realized with closed-form expressions, preserving per-cell separability and computational efficiency. The power update solves a concave quadratic with respect to $\sqrt{p_k}$; the scheduling metric naturally generalizes the classical FP approach, but with SINR-adaptive terms.

Theoretical convergence is guaranteed: each iteration yields a non-decreasing WSR and the algorithm halts upon reaching a stationary point or exceeding a maximum number of iterations. Computationally, SEFP has the same per-iteration asymptotic complexity as classical FP.

Numerical Results and Empirical Observations

Extensive simulation studies were conducted under three WSR utility metrics (Proportional-Fairness, Equal-Weight Sum Rate, and Normalized Random-Priority WSR) and across a wide SNR range. Key observations include:

• Consistent Outperformance: SEFP surpasses the classical FP and representative baselines (power control-based, fixed interference approximation) in all tested metrics.
• Strong Numerical Gains: For Proportional Fairness, SEFP achieves a statistically significant and consistent improvement in mean utility. Under random-prioritized and equal-weight metrics, SEFP overwhelmingly achieves top performance in almost all Monte Carlo realizations.
• Statistical Robustness: No single channel realization favored the classical FP over SEFP.
• Resilience Across SNR: The performance advantage persists uniformly over 0–14 dB SNR. Gains are especially manifest at moderate-to-high SNR, where intercell interference becomes the dominant performance bottleneck.

Implications and Future Directions

The introduction of RIT within fractional programming for joint uplink scheduling and power control demonstrates that algorithmic tightness at the surrogate-function level yields measurable improvements in both practical and theoretical performance. The approach is fully compatible with distributed, per-cell implementation and retains the closed-form update properties central to FP’s success in wireless resource allocation.

The theoretical advancement provided by RIT has broader implications for mixed-integer continuous optimization problems, particularly those involving log-ratio objectives common in wireless and networked systems. The nonrestrictive surrogate construction enabled by RIT may generalize to other forms of resource allocation, multiantenna (MIMO) beamforming, and potentially to signal processing problems in integrated sensing-communication or beyond.

Given SEFP’s empirical superiority and theoretical provability, future work should extend RIT-based FP to MIMO settings, examine applicability to downlink/user fairness models, and further explore acceleration strategies or learning-assisted optimization paradigms that could scale to more dynamic or heterogeneous network scenarios.

Conclusion

This work presents a theoretically motivated and empirically validated advancement of fractional programming for joint uplink scheduling and power control in multicell wireless networks. By replacing the LDT with the RIT minorization, the resulting SEFP algorithm achieves systematically tighter surrogates, leading to uniformly improved numerical performance while preserving computational tractability and per-cell separability. The findings highlight the importance of surrogate tightness in mixed discrete-continuous wireless optimization and open new directions for surrogate-based algorithmic design in broader communications and signal processing domains.