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Achieving maximum energy-efficiency in multi-relay OFDMA cellular networks: a fractional programming approach
Kent Tsz Kan Cheung, Shaoshi Yang, Lajos Hanzo
TL;DR
The paper asks how to jointly allocate power and subcarriers to maximize energy efficiency in multi-relay, multi-user OFDMA networks. It formulates the problem as quasi-concave fractional programming and solves it with Dinkelbach’s method plus dual decomposition. Results characterize the trade-off between energy and spectral efficiency across power and network configurations, while the study assumes negligible inter-cell interference.
Problem
The paper addresses energy-efficiency maximization when power and subcarrier allocation must be jointly optimized in a multi-relay, multi-user OFDMA network, unlike prior formulations focused directly on spectral efficiency or power minimization.
Method
The paper relaxes the allocation problem, proves quasi-concavity, and solves the resulting fractional program with Dinkelbach’s method and dual decomposition.
Results
EEM and SEM coincide when available power is insufficient for maximum energy efficiency; at higher power, SEM increases spectral efficiency while EEM maintains maximum energy efficiency and reaches an upper spectral-efficiency limit.
Takeaways & Limitations
The framework supports network-design analysis of how relay count and placement, subcarrier availability, user count, and power budgets affect spectral and energy efficiency.
Takeaways & Limitations
The study assumes inter-cell interference is sufficiently low to be ignored and identifies multi-cell, interference-limited systems as future work.
Abstract
from arXiv · showhide
In this paper, the joint power and subcarrier allocation problem is solved in the context of maximizing the energy-efficiency (EE) of a multi-user, multi-relay orthogonal frequency division multiple access (OFDMA) cellular network, where the objective function is formulated as the ratio of the spectral-efficiency (SE) over the total power dissipation. It is proven that the fractional programming problem considered is quasi-concave so that Dinkelbach's method may be employed for finding the optimal solution at a low complexity. This method solves the above-mentioned master problem by solving a series of parameterized concave secondary problems. These secondary problems are solved using a dual decomposition approach, where each secondary problem is further decomposed into a number of similar subproblems. The impact of various system parameters on the attainable EE and SE of the system employing both EE maximization (EEM) and SE maximization (SEM) algorithms is characterized. In particular, it is observed that increasing the number of relays for a range of cell sizes, although marginally increases the attainable SE, reduces the EE significantly. It is noted that the highest SE and EE are achieved, when the relays are placed closer to the BS to take advantage of the resultant line-of-sight link. Furthermore, increasing both the number of available subcarriers and the number of active user equipment (UE) increases both the EE and the total SE of the system as a benefit of the increased frequency and multi-user diversity, respectively. Finally, it is demonstrated that as expected, increasing the available power tends to improve the SE, when using the SEM algorithm. By contrast, given a sufficiently high available power, the EEM algorithm attains the maximum achievable EE and a suboptimal SE.
I. INTRODUCTION
The paper addresses energy-efficiency maximization in multi-relay, multi-user OFDMA networks by jointly optimizing power and subcarrier allocation. It develops an efficient solution framework and compares energy-efficiency and spectral-efficiency objectives for network design.
- Prior spectral-efficiency and power-minimization formulations do not directly optimize the energy-efficiency objective.
- Energy-efficiency maximization jointly optimizes power and subcarrier allocation in a multi-relay, multi-user OFDMA cellular network.
- The framework combines variable transformation and relaxed subcarrier indicators to make the original mixed-integer nonlinear problem more tractable.
- Dinkelbach’s method and dual decomposition obtain the optimal solution with low complexity after the relaxed problem is shown to be quasi-concave.
- As available power increases, SEM seeks higher spectral efficiency at lower energy efficiency, whereas EEM preserves maximum energy efficiency and reaches an upper spectral-efficiency limit.
- The generalized model supports studying how subcarriers, users, relays, cell radius, and relay positions affect attainable spectral and energy efficiency.
III. PROBLEM FORMULATION
The problem formulation maximizes energy efficiency, expressed as a spectral-efficiency-to-power ratio, under allocation, protocol-selection, and total-power constraints. Relaxing binary subcarrier indicators enables a tractable approximation whose dual approaches the original problem as the number of subcarriers grows.
- The objective maximizes energy efficiency as the ratio between spectral efficiency and total power dissipation under a maximum total instantaneous transmit-power constraint.
- The formulation contains continuous direct and amplify-and-forward power variables together with binary subcarrier-indicator variables, making it a mixed-integer nonlinear problem.
- Relaxing binary subcarrier indicators to values in [0, 1] permits time-sharing allocation between user-subcarrier pairs.
- Dual solutions to the relaxed problem can approach the original non-relaxed problem arbitrarily closely as the number of available subcarriers tends to infinity.
- The relaxed formulation enforces the power budget, selects one transmission protocol per user-subcarrier pair, and assigns each subcarrier to at most one user.
A. Proving that the OF in problem (P) is quasi-concave
The energy-efficiency objective is proven quasi-concave by showing a concave numerator, an affine positive denominator, and a convex feasible domain. Concavity follows from the logarithmic structure, perspective transformations, and nonnegative sums of concave terms.
- Quasi-concavity follows because the objective has a concave numerator, an affine positive denominator, and a convex domain.
- The complete numerator remains concave because it is a nonnegative sum of concave component functions.
- The direct-transmission term is shown concave through a negative-semidefinite Hessian with non-positive eigenvalues.
- The logarithmic composition is concave because log2(·) is concave and non-decreasing over the relevant argument.
- A perspective transformation preserves concavity for the amplify-and-forward contribution.
B. Problem solution methods
Because quasi-concavity does not make standard convex optimization directly applicable, the paper uses iterative methods to approach the optimum through parameterized concave problems.
- Quasi-concavity provides convex superlevel sets but does not guarantee that a local maximum is global, limiting direct use of standard convex optimization methods.
A. Introduction to Dinkelbach’s method
Dinkelbach’s method transforms the quasi-concave energy-efficiency problem into a sequence of parameterized concave problems. Dual decomposition then solves each secondary problem through similar subproblems and a dual-variable update.
- Method: Dinkelbach’s method solves the quasi-concave fractional problem through iterative parameterized concave optimization.The method uses the subtractive objective F(q) = R_T(P, S) − qP_T(P, S).
- Method: At the optimal q*, maximizing F(q*) is equivalent to solving the original fractional problem.The optimality condition is expressed through the vanishing subtractive objective at q*.
- Convergence: The algorithm generates increasing q values that converge superlinearly to the optimal value.Each outer iteration solves a parameterized problem using the previous q value.
- Concavity and duality: Each parameterized problem is concave because spectral efficiency is concave and total power is affine.Under Slater’s condition, the associated dual and primal problems have zero duality gap.
- Decomposition: Dual decomposition solves each secondary problem through NK similar subproblems and a master problem that updates λ.The Lagrangian incorporates λ for the total-power constraint.
1) Solving the subproblem of power and subcarrier allocation:
For a fixed dual variable, the allocation subproblem is solved using KKT conditions, customized water-filling for power, and derivative-based subcarrier assignment. The dual variable is then updated iteratively until convergence.
- Subproblem solution: KKT conditions provide the optimal power and subcarrier allocations after the subproblem is expressed in standard concave form.The optimal variables are derived for a given dual variable λ.
- Power allocation: The optimal direct and relayed powers are obtained from first-order derivatives under nonnegative-power constraints.The nonnegative-power requirement is enforced through max(0, ·).
- Power allocation: The relay power split is constrained by 0 ≤ β_A,n ≤ 1.β_A,n represents the fraction of AF transmit power allocated to the BS-to-relay link.
- Subcarrier allocation: Each subcarrier is assigned to the user producing the largest increase in the Lagrangian.This follows because each subcarrier may serve only one user.
- Power allocation: The resulting power allocations are customized water-filling solutions whose water levels depend on λ and the current energy-efficiency parameter.Effective channel gains determine the allocation alongside these power costs.
- Dual update: The dual variable λ is updated by a gradient method and allocation updates repeat until the dual optimum is reached.The step size α_λ(i) controls the update at iteration i.
C. Summary of solution methodology
The relaxed fractional problem is solved through nested Dinkelbach outer iterations and dual-decomposition inner iterations, yielding optimal power and subcarrier allocations at complexity O(Idual × 2NK).
- Outer iterations: Each Dinkelbach outer iteration rewrites the fractional problem as a parameterized subtractive concave problem for a given qi.The resulting secondary problem is solved by dual decomposition.
- Inner iterations: Each inner iteration solves 2NK subproblems for the power and subcarrier variables, then updates the dual variable λ.Inner iterations continue until the primal and dual solutions converge.
- Convergence: The converged power and subcarrier allocations are fed back to update qi, and outer iterations continue until qi converges.The resulting P* and S* solve the relaxed problem (P).
- Complexity: O(Idual × 2NK) is the total complexity when NK is large, dominated by comparison operations in the inner-loop solution.Idual denotes the total number of inner iterations required for convergence in Dinkelbach’s method.
V. RESULTS AND DISCUSSIONS
The results evaluate the relay-aided cellular system using average SE and EE per subcarrier, with ρ measuring the fraction of subcarriers used for AF transmission.
- Simulation setup: The simulations use path-loss and uncorrelated Rayleigh fading, with LOS propagation assumed for BS-to-RN links.BS-to-UE and RN-to-UE links are typically modeled without LOS because of blockage.
- Simulation setup: Average SE and EE per subcarrier are used for fair comparisons, while sum-rate equals average SE multiplied by NW.ρ denotes the average fraction of subcarriers used for AF transmission.
- Simulation setup: RNs are evenly distributed at a fixed distance around the central BS, and UEs are uniformly distributed within the cell.Independent UE locations and fading realizations are generated for each channel sample.
A. Convergence of iterative algorithms to optimal value
The iterative EEM procedure converges to the exhaustive-search optimum in small systems, while the reported design analysis compares SE, EE, and relay-use behavior under changing system parameters.
- Convergence: Forty inner iterations suffice for Dinkelbach’s method to converge to the optimal EE value in the evaluated small-scale systems.The result is averaged over 10^4 channel realizations and compared with exhaustive search.
- Convergence: The EEM algorithm obtains the optimal power and subcarrier allocation despite solving the relaxed problem and assuming a high receiver SNR.The reported optimum is the exhaustive-search-based solution for the evaluated setting.
B. Effect of the number of UEs on the attainable SE and EE
The paper examines how user count and other system parameters affect attainable spectral efficiency (SE), energy efficiency (EE), and relay usage. More users increase both SE and EE through multi-user diversity, while subcarrier count, cell radius, and relay placement produce distinct trade-offs.
- Increasing K raises both maximum EE and attained SE through greater multi-user diversity.The scheduler can select subcarrier allocations from a larger pool of channel gains.
- As Pmax increases, SEM raises SE at the cost of EE, whereas EEM maintains maximum EE with corresponding SE.The EEM and SEM trends are reported for the increasing-subcarrier analysis.
- Increasing N raises SEM sum-rate through frequency diversity but decreases average SE because not all subcarriers are effectively utilized.The reported average SE is averaged over N.
- Increasing N raises ρ because the scheduler accesses more channel gains for each UE and can support more cell-edge users.This contrasts with increasing K, which favors nearer-to-center UEs and lowers ρ.
- Increasing cell radius harms both SE and EE through increased pathlosses, while increasing ρ indicates greater relay benefit in larger cells.With M = 6 instead of M = 0 at 2 km, SE improves by a factor of 1.03 but EE changes by a factor of 0.34.
- Optimal SE and EE occur when relays are closer to the BS than to UEs, because a stronger BS-to-relay LOS link strengthens AF links.Placing relays too close to the BS makes the RN-to-UE link more hostile.
VI. CONCLUSIONS
The paper formulates joint power and subcarrier allocation for EE maximization and solves it using Dinkelbach’s method with dual decomposition. Simulations characterize EEM and SEM trade-offs and show how users, subcarriers, relays, cell size, and relay placement affect SE and EE.
- VI. CONCLUSIONS: The relaxed EE maximization problem is quasi-concave, enabling Dinkelbach’s method to solve parameterized concave problems using dual decomposition.The algorithm reaches the exhaustive-search optimum within a low number of iterations.
- VI. CONCLUSIONS: When available power is insufficient for maximum EE, EEM and SEM have the same solution; with more power, SEM increases SE while EEM reaches an upper bound.EEM does not use additional available power after reaching its EE limit.
- VI. CONCLUSIONS: More UEs increase SE and EE through multi-user diversity, whereas more subcarriers increase sum-rate through frequency diversity but reduce average SE.The conclusion attributes the subcarrier result to ineffective utilization of all available subcarriers.
- VI. CONCLUSIONS: Adding relays provides marginal SE benefit but degrades EE because additional transmitting entities increase overhead power consumption.Relay benefits become more significant with larger cells or more relays.
- VI. CONCLUSIONS: Relaying is more beneficial when relays are closer to the BS and a LOS link exists between the relays and BS.
- VI. CONCLUSIONS: The study assumes inter-cell interference is sufficiently low to be ignored, unlike the interference-limited multi-cell systems proposed as future work.Online near-real-time optimization for mobile relays is also identified as a possible next step.