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Energy-Efficient Power Control: A Look at 5G Wireless Technologies

Alessio Zappone, Luca Sanguinetti, Giacomo Bacci, Eduard Jorswieck, Mérouane Debbah

arXiv:1503.04609v2cs.ITmath.OC

TL;DR

The paper addresses energy-efficient power control under minimum-rate constraints and generalized SINR models covering emerging 5G technologies. It combines fractional programming, sequential convex optimization, and game theory to design centralized and decentralized algorithms. The methods converge to KKT points or generalized Nash equilibria, while the gap to global optima remains an open challenge.

  • Problem

    Energy-efficiency optimization is difficult when minimum-rate constraints and interference make the relevant fractional problems non-convex.

  • Method

    The paper combines fractional programming, sequential convex optimization, and game theory to develop centralized and fully decentralized EE power-control algorithms.

  • Results

    The centralized algorithms converge to KKT points, while the decentralized best-response algorithms converge to generalized Nash equilibria under the analyzed conditions.

  • Takeaways & Limitations

    The framework jointly supports network-centric and user-centric EE optimization with rate constraints across single or multiple resource blocks.

  • Takeaways & Limitations

    The gap between the proposed centralized solutions and the global optima of GEE and minimum EE remains unresolved.

Abstract

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This work develops power control algorithms for energy efficiency (EE) maximization (measured in bit/Joule) in wireless networks. Unlike previous related works, minimum-rate constraints are imposed and the signal-to-interference-plus-noise ratio takes a more general expression, which allows one to encompass some of the most promising 5G candidate technologies. Both network-centric and user-centric EE maximizations are considered. In the network-centric scenario, the maximization of the global EE and the minimum EE of the network are performed. Unlike previous contributions, we develop centralized algorithms that are guaranteed to converge, with affordable computational complexity, to a Karush-Kuhn-Tucker point of the considered non-convex optimization problems. Moreover, closed-form feasibility conditions are derived. In the user-centric scenario, game theory is used to study the equilibria of the network and to derive convergent power control algorithms, which can be implemented in a fully decentralized fashion. Both scenarios above are studied under the assumption that single or multiple resource blocks are employed for data transmission. Numerical results assess the performance of the proposed solutions, analyzing the impact of minimum-rate constraints, and comparing the network-centric and user-centric approaches.

I. INTRODUCTION

The paper targets energy-efficient power control for 5G wireless networks under minimum-rate constraints and a generalized SINR model. It develops centralized and decentralized approaches for network-centric and user-centric EE objectives across single or multiple resource blocks.

  • Motivation: 5% of global CO2 emissions are estimated to come from ICT, while connected devices and required data rates are projected to increase sharply.The paper argues that meeting a 1000x data-rate increase by scaling transmit power would create unmanageable energy demand and emissions.
  • Framework: The framework covers K transmitters sharing N mutually orthogonal resource blocks and supports both centralized and decentralized power allocation.Network-centric methods coordinate nodes around a common objective, whereas user-centric methods operate distributively.
  • Model: Minimum-rate constraints and a generalized SINR expression extend the framework to emerging 5G technologies, including hardware-impaired massive MIMO and imperfect CSI settings.The generalized expression also encompasses heterogeneous and relay-assisted interference networks.
  • Objectives: The paper maximizes GEE and minimum EE centrally, while modeling individual EE maximization as a decentralized non-cooperative game.The decentralized formulation studies equilibrium existence and uniqueness and uses best-response dynamics to reach equilibrium.
  • Algorithms: Centralized algorithms combine fractional programming with sequential convex optimization to converge to KKT points of the non-convex EE problems.The approach is designed for affordable computational complexity and includes closed-form feasibility conditions.
  • Evaluation scope: The study evaluates single- and multi-resource-block transmission scenarios and compares network-centric with user-centric EE optimization.The formulation includes individual power vectors, maximum-power constraints, and minimum achievable-rate requirements.

A. Network-centric formulation

The network-centric formulation evaluates system-wide benefit-cost efficiency and fairness-oriented minimum user EE. These non-convex fractional objectives are paired with a game-theoretic user-centric formulation for individual EE optimization.

  • A. Network-centric formulation: The network GEE is defined as the system achievable sum-rate divided by total consumed power.The total power includes network circuit power and users’ transmit powers.
  • A. Network-centric formulation: The weighted minimum-EE metric maximizes the smallest weighted user EE across the network.The weights allow the allocation to reflect different fairness preferences among users.
  • A. Network-centric formulation: Both network-centric optimization problems are non-convex fractional programs addressed using fractional programming and sequential convex optimization.The paper presents these tools as the solution framework for the GEE and weighted minimum-EE formulations.
  • A. Network-centric formulation: GEE emphasizes global benefit-cost performance, whereas weighted minimum EE provides a fairer allocation across users.Maximizing weighted minimum EE yields a Pareto-efficient point, while varying weights traces the Pareto boundary with possible system-efficiency loss.
  • B. User-centric formulation: The user-centric power-allocation problem is formulated for a game-theoretic solution in which users optimize individual EE.The formulation can also accommodate a weighted sum-rate definition of GEE.
  • B. User-centric formulation: Fractional programming is introduced as the relevant background for ratio optimization in the formulation.The paper provides additional background in Appendix A.
  • B. User-centric formulation: The formulation defines a maximum achievable SINR by considering the absence of interference and thermal noise, then rewrites the general SINR accordingly.The resulting SINR is strictly increasing in each user’s resource-block transmit power.

C. Applications to 5G technologies

The general SINR framework covers massive MIMO and relay-assisted interference networks, including hardware impairments and amplify-and-forward relaying. These application models produce SINR expressions of the paper’s general form, though receiver choices constrain applicability.

  • Massive MIMO: Massive MIMO uplink SINR can be written in the paper’s general form using channel combining and a lower-bound analysis.The model considers multi-cell systems with multi-antenna base stations and single-antenna user equipments.
  • Massive MIMO: Hardware impairments introduce signal reduction and additive distortion noise, while the resulting SINR retains the same form.The analysis is extended to impairments at both user equipments and base stations.
  • Relay-assisted CoMP interference network: Relay-assisted multi-point interference networks use amplify-and-forward relaying, with normalization before amplification and forwarding to receivers.After linear reception, the resulting SINR takes the general form in (1).
  • Relay-assisted CoMP interference network: The framework applies to relay-assisted multi-cell, small-cell, and relay-assisted device-to-device networks.Its applicability follows because the received-signal model produces the same general SINR structure.
  • Receiver dependence: Receive filters affect SINR coefficients and equivalent noise power; power-independent filters support the framework, including MRC and zero-forcing but not readily MMSE receivers.MMSE receivers depend on transmit powers through the interference covariance matrix.

A SINGLE RESOURCE BLOCK

With one resource block, the power-control problems become analytically more tractable than their multi-resource-block counterparts. This setting permits necessary-and-sufficient feasibility analysis under minimum-rate constraints.

  • Single-resource-block setting: N = 1 models single-carrier systems and provides deeper analytical insight than the N > 1 case.The single-resource-block setting is explicitly described as more analytically tractable.
  • Feasibility: The single-resource-block case permits necessary and sufficient feasibility conditions for centralized energy-efficient optimization problems.For multiple resource blocks, only sufficient feasibility conditions are obtained.
  • Feasible power sets: Each user’s feasible power set imposes both a maximum-power constraint and a minimum-rate constraint.The set is written as P_k = {p_k ∈ R_+ : p_k ≤ p̄_k, log2(1 + γ_k) ≥ θ_k}.

A. Feasibility

The feasibility analysis characterizes when minimum-rate and power constraints admit a solution, then develops an efficient KKT-convergent method for global EE maximization’s non-convex objective.

  • Feasibility: Closed-form necessary and sufficient conditions determine whether the feasible power set is non-empty.The conditions are expressed using a nonnegative matrix F and its spectral radius ρ_F.
  • Feasibility: If ρ_F ≥ 1, no nonnegative power vector can satisfy the target SINR requirements.The necessity argument uses the monotonicity of each user’s SINR in its own power.
  • GEE maximization: Fractional programming alone is insufficient because the GEE objective lacks a concave numerator under interference.The paper combines fractional programming with sequential convex programming to address this non-convexity.
  • GEE maximization: The transformed subproblem has a concave numerator, convex denominator, and convex feasible set, enabling efficient fractional optimization.The logarithmic change of variables is used to establish these properties.
  • GEE maximization: The iterative centralized algorithm monotonically increases GEE and converges to a point satisfying the original problem’s KKT conditions.Each iteration solves a tractable approximating problem after lower-bounding the logarithmic rate expression.

C. Weighted Minimum EE Maximization

Minimum EE maximization involves multiple fractional utilities, requiring generalized fractional programming alongside sequential convex optimization. The resulting algorithm converges to KKT points, while the decentralized formulation studies equilibrium under coupled strategies.

  • Weighted Minimum EE Maximization: The minimum-EE objective contains K fractional functions, so ordinary Dinkelbach’s algorithm does not apply directly.Generalized Dinkelbach’s algorithm handles the resulting max-min fractional problem when ratio curvature conditions hold.
  • Weighted Minimum EE Maximization: The proposed procedure monotonically increases minimum EE and converges to a KKT point of the epigraph-form problem.Sequential convex optimization is combined with generalized Dinkelbach iterations.
  • Distributed power control: The distributed formulation models users as non-cooperative agents maximizing individual EE subject to power and rate constraints.Each user’s feasible strategy set depends on the other users’ powers.
  • Distributed power control: A best-response fixed point is a Nash equilibrium, but coupled strategy sets make existence, uniqueness, and convergence more restrictive questions.The paper treats this setting as a generalized non-cooperative game.

A. Analysis of the Equilibria

The distributed EE game has equilibria under sufficient feasibility conditions, with uniqueness and best-response convergence established for single and multiple resource blocks. The resulting iterative procedures can be implemented fully decentralized using locally measured SINR information.

  • The game admits a nonempty set of generalized Nash equilibrium points under sufficient feasibility conditions.
  • The feasibility assumptions ensure nonempty, convex, closed, and bounded strategy sets that vary continuously with other players’ powers.User payoffs are quasi-concave because they are strictly pseudo-concave ratios of a strictly concave numerator and an affine denominator.
  • The game admits a unique GNE, and best-response dynamics converge to it from any feasible power vector.
  • Single resource block: The single-resource-block algorithm updates each user’s power through best responses and Dinkelbach’s algorithm until convergence.The procedure initializes feasible powers, receives SINR feedback, computes equivalent channel gain, and updates power iteratively.
  • Single resource block: Only the equivalent channel gain is needed for each update, and it can be obtained from locally measured SINR returned over a downlink channel.Therefore, the algorithm both converges to the unique GNE and supports fully decentralized implementation.
  • Multiple resource blocks: For multiple resource blocks, non-convex rate constraints permit sufficient feasibility conditions and low-complexity algorithms converging to KKT points.

A. GEE maximization

For global EE maximization with multiple resource blocks, the paper lower-bounds the non-convex rate constraints and solves successive convex subproblems with Dinkelbach’s algorithm. The resulting procedure monotonically improves global EE and converges to a KKT point.

  • The multi-resource-block method lower-bounds each non-convex rate constraint with a concave function after the logarithmic power transformation q_k,n = log2 p_k,n.
  • The resulting power-allocation subproblem is solved using Dinkelbach’s algorithm within Algorithm 3.
  • Algorithm 3 monotonically increases the GEE value and converges to a point satisfying the KKT conditions of the original non-convex problem.
  • Algorithm 3 initializes a feasible power point, constructs initial approximations, and iteratively solves the GEE and minimum-EE subproblems until convergence.

B. Weighted minimum EE maximization

The weighted minimum-EE problem is handled with a concave-over-convex reformulation and Generalized Dinkelbach’s algorithm. The resulting procedure has a sufficient feasibility test and, for multiple resource blocks, supports unique-equilibrium distributed power control.

  • The weighted minimum-EE objective is lower-bounded using the increasing min(·) function and the same concave approximation used for global EE.
  • The reformulated problem has a concave numerator and convex denominator, enabling global solution with the Generalized Dinkelbach’s algorithm.
  • Algorithm 3 monotonically increases the minimum-EE value and converges to a KKT point of the epigraph-form original non-convex problem.
  • Feasibility of the original global-EE and minimum-EE problems is sufficient when their convex surrogate tests are feasible, and this feasibility persists across iterations.
  • Multiple resource blocks: For multiple resource blocks, a unique GNE exists and best-response dynamics converge whenever the stated sufficient condition holds for every user.
  • Multiple resource blocks: The multiple-resource-block iterative procedure computes per-block equivalent gains from measured SINR and converges to the unique GNE in a fully decentralized fashion.

VII. NUMERICAL RESULTS

The numerical evaluation considers two case studies: a hardware-impaired massive MIMO system and a multi-carrier relay-assisted interference network.

  • The proposed solutions are evaluated in a hardware-impaired massive MIMO system and a multi-carrier relay-assisted interference network.

A. Hardware-Impaired Massive MIMO System

The evaluation examines feasibility, energy efficiency, convergence, and rate trade-offs for centralized and distributed power-control algorithms in massive MIMO and relay-assisted networks. Minimum-rate constraints improve users’ rates but can reduce GEE, with the impact depending on architecture and resource blocks.

  • Massive MIMO feasibility: Feasibility probability approaches 1 for realistic transmit powers up to R = 25%, whereas R = 30% is not reliably feasible at typical uplink power levels.The estimate averages over 5·10^4 independent user-drop and channel-coefficient scenarios.
  • Massive MIMO performance: In the massive-MIMO saturation region, minimum rate increases from 1.6 [bit/s/Hz/user] without QoS constraints to 2.35 [bit/s/Hz/user] with R = 20%.The QoS-constrained scheme uses some excess power to satisfy minimum-rate requirements while maintaining near-peak GEE.
  • Massive MIMO performance: Minimum-EE maximization exhibits behavior similar to GEE maximization when comparing constrained, unconstrained, and min-rate-maximizing policies.Figure 3 evaluates average minimum users’ energy efficiency versus P for R = 20% and R = 0%.
  • Centralized versus distributed allocation: Introducing QoS constraints causes a larger GEE degradation in distributed resource allocation than in centralized optimization, especially as P increases.The distributed scheme does not jointly manage user interference, producing higher multi-user interference at large P.
  • Convergence: Both Algorithms 1 and 2 converge after few iterations, with the distributed algorithm converging faster and therefore suiting self-organizing networks.The iteration count increases slightly with P because larger P produces a larger feasible set.
  • Relay-assisted OFDMA: In the relay-assisted OFDMA scenario, minimum rate increases from 3.8 [bit/s/Hz/user] at R = 0% to 7.16 [bit/s/Hz/user] at R = 20%, with a slight GEE reduction.The centralized–decentralized GEE gap is rather limited, potentially reflecting channel diversity from multiple sub-carriers and the limited number of non-cooperating transmitters.

C. Computational complexity discussion

The proposed centralized and distributed algorithms have polynomial per-stage complexity, with limited outer iterations in both single- and multi-resource-block settings. Centralized methods perform better under rate constraints but require greater computational complexity and feedback, while the framework remains short of global-optimality guarantees.

  • Iteration complexity: Limited outer iterations are observed for both centralized and distributed algorithms in single- and multi-resource-block settings.The distributed algorithms require slightly fewer outer iterations in the single-resource-block case.
  • Centralized algorithms: The centralized algorithms have polynomial complexity in each stage, making their computational complexity affordable.The auxiliary problems have polynomial complexity in the number of variables and constraints.
  • Distributed algorithms: Distributed algorithms also have polynomial per-stage complexity, with each stage depending on resource blocks and constraints rather than the number of users.Each outer-loop stage solves K fractional problems, each with N variables and two constraints.
  • Performance trade-offs: Centralized algorithms are more robust to demanding rate constraints and outperform distributed methods, at the cost of higher computational complexity and feedback requirements.Distributed algorithms become more sensitive as maximum feasible powers increase because they lack centralized interference management.
  • Open limitation: The gap between the proposed centralized solutions and the global optima of GEE and minimum EE remains an open research problem.Determining global solutions in interference-limited networks is identified as challenging.
  • Centralized algorithms: Dinkelbach’s algorithm solves one convex auxiliary problem per iteration and converges super-linearly when the numerator is concave and denominator convex.This supports polynomial complexity for each stage of the centralized algorithms.

APPENDIX B

The appendix reviews convergence mechanisms for fractional and sequential optimization procedures. It establishes monotonic improvement and KKT convergence for the relevant approximations, while also stating a uniqueness condition for the generalized Nash equilibrium.

  • Sequential optimization: Each sequential-convexification iteration maximizes a lower bound subject to the original constraints, producing a nondecreasing objective sequence.The objective increases because the bound is tight at the previous iterate, and convergence follows from an upper bound.
  • Sequential optimization: The converged lower-bound solution satisfies the original problem’s KKT conditions because the bound and objective coincide there with matching gradients.This connects the approximate optimization problem to first-order stationarity of the original formulation.
  • Constrained optimization: The modified algorithm for lower-bounding a constraint function converges to a point fulfilling the KKT conditions of the constrained problem.The approximate solution remains feasible because the original constraint function dominates its lower bound.
  • Game-theoretic analysis: The generalized Nash equilibrium is unique when the stated interference-response contraction condition is below one for every user.The condition is expressed using the supremum norm of the relevant interference derivatives.
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