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A Minorization-Maximization Method for Optimizing Sum Rate in Non-Orthogonal Multiple Access Systems
Muhammad Fainan Hanif, Zhiguo Ding, Tharmalingam Ratnarajah, George K. Karagiannidis
TL;DR
The paper studies downlink sum-rate maximization for linearly precoded MISO NOMA, where the underlying optimization is non-convex. It applies an MMA that solves SOCP subproblems, reports few-iteration convergence and polynomial complexity, and develops a reduced-complexity approximation whose tightness conditions are examined.
Problem
The paper addresses downlink sum-rate maximization in a MISO system using NOMA principles, with the goal of maximizing total throughput while satisfying NOMA constraints.
Method
The proposed MMA approximates the original problem through iterative SOCP-based optimization and also develops a reduced-complexity approximation.
Results
NOMA is reported to outperform conventional orthogonal multiple-access schemes, with high data rates at small transmit power.
Takeaways & Limitations
The algorithm converges within a few iterations, while the reduced-complexity approximation is studied through its tightness conditions and occurrence probability.
Abstract
from arXiv · showhide
Non-orthogonal multiple access (NOMA) systems have the potential to deliver higher system throughput, compared to contemporary orthogonal multiple access techniques. For a linearly precoded multiple-input multiple-output (MISO) system, we study the downlink sum rate maximization problem, when the NOMA principles are applied. Being a non-convex and intractable optimization problem,we resort to approximate it with a minorization-maximization algorithm (MMA), which is a widely used tool in statistics. In each step of the MMA, we solve a second-order cone program, such that the feasibility set in each step contains that of the previous one, and is always guaranteed to be a subset of the feasibility set of the original problem. It should be noted that the algorithm takes a few iterations to converge. Furthermore, we study the conditions under which the achievable rates maximization can be further simplified to a low complexity design problem, and we compute the probability of occurrence of this event. Numerical examples are conducted to show a comparison of the proposed approach against conventional multiple access systems. NOMA is reported to provide better spectral and power efficiency with a polynomial time computational complexity.
I. INTRODUCTION
The introduction motivates NOMA as a power-domain multiple-access approach for improving throughput and resource efficiency, then frames this paper’s MISO sum-rate optimization problem and MMA-based solution.
- NOMA multiplexes additional users in the same time, frequency, or code slot using the power domain.
- NOMA superposes users’ messages, orders users by effective channel gains, and allocates a higher fraction of power to weaker users.
- Sharing a channel slot is expected to improve sum rates, while intelligent power allocation can support spectral efficiency, reliability, quality of service, and weaker users.
- Prior work: Prior work studied NOMA mainly in SISO or limited MIMO settings, including outage, fairness, SIC error propagation, and fixed power allocation.
- Problem focus: This paper designs complex weighting vectors for MISO downlink sum-rate maximization under a given user ordering rather than solving the optimal ordering problem.
- Contributions: The proposed MMA iteratively solves an SOCP with decodability and non-zero-rate constraints, replacing traditional SDP-based approaches.
- Contributions: The algorithm is reported to converge in few iterations, has polynomial worst-case complexity, and under plausible assumptions converges to a KKT point.
- Numerical findings: Numerical results report that NOMA outperforms orthogonal access particularly at low SNR and when users outnumber BS antennas, while distance affects throughput when the approximation is exact.
C. Structure
This section outlines the paper’s organization and notation, then describes the MISO NOMA system, signal model, SIC operation, and fixed-ordering design scope.
- Organization: The paper proceeds from system modeling and preliminaries to algorithm development, reduced-complexity approximation, numerical results, and conclusions.
- System model: The downlink system has a BS with T antennas serving N single-antenna users under NOMA transmission.
- Signal model: Each user’s symbol is multiplied by a complex precoding vector, and the BS transmits the superposition of all users’ weighted messages.
- Signal model: Under frequency-flat channels, UE-i receives its channel-weighted superposition signal plus complex Gaussian noise.
- SIC: SIC follows a specified user ordering: UE-k decodes selected other-user signals, while Fig. 1 shows interference from later users and cancellation of earlier users.
- Scope: The study does not transform simple SISO ordering to MISO or optimize ordering; it optimizes weighting vectors for a given ordering with perfect CSI assumed.
A. Problem Formulation
The paper formulates downlink NOMA sum-rate maximization for a linearly precoded MISO system, accounting for direct and cross-user decoding requirements under an ordered user structure.
- UE-1 is assumed to be the weakest user and therefore cannot decode interfering signals.
- Users are ordered by increasing index, although ordering by channel strength may not be optimal and alternative orders may achieve better rates.
- Higher-ranked users must satisfy decoding-rate constraints for messages from weaker users to enable successive interference cancellation.
- The formulation maximizes sum rate while enforcing ordered received-signal strengths and a total transmit-power bound Pth.
III. PREREQUISITES
The original formulation is transformed into an equivalent representation that exposes its non-convex constraints while expressing the objective through auxiliary rate variables and second-order cone constraints.
- The original optimization problem is transformed before applying iterative approximations to improve tractability.
- The transformed problem remains intractable because several constraints are non-convex, although the remaining constraints are convex and admit SOC representation.
- The reformulation introduces auxiliary variables for rates and decoding constraints while preserving equivalent relationships among SINR terms.
- The geometric mean of the rate vector is concave and increasing, and can be represented using second-order cone constraints.
- The non-convex constraints are handled separately after factoring the formulation into distinct constraint groups.
B. Approximation of the non-convex constraints
The paper approximates the non-convex constraints using CCP/MMA linearizations, producing convex surrogate constraints suitable for iterative optimization.
- The convex-concave procedure, also called minorization-maximization, is used to handle the non-convex constraints.
- The same approximation strategy is applied successively to the non-convex constraint sets in (14b), (14c), (14d), and (14e).
- Bilinear and quadratic terms are replaced by first-order Taylor approximations around values from the current iteration.
- The approximation preserves a minorant-based iterative structure in which the current point defines the next surrogate iterate.
- The resulting constraint involving the approximated terms is convex in the variables of interest.
IV. THE PROPOSED SOLUTION
The proposed MMA algorithm solves a sequence of SOCPs with nested feasibility properties, monotonic objective improvement, and convergence to a KKT point under stated conditions.
- The Proposed Solution: Each MMA iteration solves an optimization problem constrained by the current convexified formulations and auxiliary-variable definitions.
- The Proposed Solution: The algorithm’s sequence of iterates remains feasible for the original problem, with each iterate belonging to the initial feasible set.
- Convergence: The objective values are non-decreasing, satisfying O_t+1 ≥ O_t, and therefore converge.
- Convergence: Under convexity and compactness of the feasible set, the algorithm converges to a finite value.
- Convergence: As t tends to infinity, the algorithm converges to a KKT point of the original formulation (14).
- Reduced Complexity Approximation: The reduced formulation omits N(N −1)/2 SINR terms and correspondingly removes N^2 −N inequality constraints.
- Reduced Complexity Approximation: When channels are clearly ordered and the stated channel-ratio inequalities hold, the reduced and original formulations are approximately equivalent.
- Reduced Complexity Approximation: The probability that the stronger-user SINR exceeds the weaker-user SINR increases as the channel-strength ratio becomes more favorable.
A. Complexity
The proposed NOMA formulations solve second-order cone programs whose worst-case complexity is determined by SOC dimensions, constraints, variables, and MMA iterations. C-NOMA and A-NOMA have polynomial-time per-iteration complexity that grows with N and T.
- Each MMA iteration solves an SOCP, and the algorithm’s worst-case complexity is determined by the SOCP solved at each step.
- General interior-point SOCP complexity depends on the numbers of constraints and variables and the dimensions of SOC constraints.
- The formulations contain 0.5N3 + 0.5N2 + 2N + c and 0.5N3 − 0.5N2 + 3N + c constraints, respectively.Here, c denotes SOC constraints with different N.
- The SOCP complexity estimates are plotted as functions of both N and T, showing how complexity increases with system dimensions.
- C-NOMA uses the SOCP in (29), whereas A-NOMA uses the SOCP in (35).
VI. NUMERICAL RESULTS
Numerical experiments evaluate C-NOMA, A-NOMA, and ZF across transmit power, user distance, and user-to-antenna ratios. The results show convergence within 25 iterations and performance changes driven by channel ordering, distance attenuation, and multiuser diversity.
- For T = N = 3, C-NOMA and A-NOMA sum rates are equal up to 25 dB transmit power because distance-based channel ordering remains valid.
- Above 25 dB, A-NOMA produces better rates by boosting the last user’s interference-free rate, while reducing the rates of the other N − 1 users.
- ZF performs poorly at lower SNRs but improves at sufficiently high transmit power as distance-induced channel ill-conditioning is partially circumvented.
- For N = T = 4, A-NOMA and C-NOMA overlap at low SNRs, whereas C-NOMA outperforms A-NOMA at higher transmit SNR.
- Decreasing D0 from 50 to 10 meters enlarges the C-NOMA/A-NOMA gap and reports higher overall data rates because users have better channel conditions.
- Reducing D0 from 50 m to 10 m considerably enhances ZF sum rates by minimizing path loss and improving channel conditioning.
- When N > T = 3, randomly selecting three users can make C-NOMA underperform because effective multiuser diversity is likely lost.
VII. CONCLUSION
The paper studies sum rate maximization for a MISO downlink using NOMA and develops approximation methods with polynomial-time iterative optimization. Numerical results report fast convergence, reduced-complexity conditions, and superior NOMA performance relative to conventional orthogonal schemes.
- The study addresses sum rate maximization in a MISO downlink system using NOMA.
- The non-convex optimization problem is approximated with a minorization-maximization method, solving an SOCP at each iteration.Each step has polynomial computational complexity.
- The proposed algorithm numerically converges within a few iterations in the considered scenarios.
- A reduced-complexity approximation is developed, together with conditions under which it is tight and an analysis of its tightness.
- NOMA has superior performance compared to conventional orthogonal multiple access schemes, obtaining high data rates with small transmit power.
- NOMA particularly outperforms ZF when the number of users is higher than the number of transmit antennas, supporting its candidacy for next-generation 5G multiple access.
APPENDIX A
The appendix establishes properties of the convex minorants and feasible sets used by the iterative approximation, then connects convergence points to KKT conditions under stated assumptions.
- The surrogate functions replacing non-convex terms are non-decreasing across iterations, yielding nested feasible sets with F_t+1 ⊇ F_t.
- Because the feasible sets expand monotonically, the objective sequence is non-decreasing and may converge to a positive value.
- The analysis assumes that the variable sequence generated by the algorithm converges to a value V∗.
- It also assumes that the constraints in the approximate problem are qualified at the limiting point.
- The appendix represents convex constraints abstractly and obtains approximated constraints by replacing non-convex functions with convex minorants.
- At convergence, the KKT conditions of the approximate problem reduce to those of the original problem and the simplified problem.
APPENDIX D
The appendix derives probability distributions relevant to the NOMA analysis under channel and precoding assumptions, using Gaussian transformations and standard distribution identities.
- For arbitrary user and node indices, the derivation uses assumptions on the noise variances and recursively related channel coefficients.
- With random unitary precoding, unitary transformations preserve the complex Gaussian distribution of the channel variables.
- The resulting x_i^k variable is exponentially distributed, while y_i^k follows a Chi-square distribution.
- The appendix derives the cumulative distribution function and probability density function of SINR_i^k from these component distributions.
- The probability calculation is completed using standard referenced distribution identities after applying the unitary Gaussian transformation.