A Minorization-Maximization Method for Optimizing Sum Rate in Non-Orthogonal Multiple Access Systems

• NOMA multiplexes additional users in the same time, frequency, or code slot using the power domain.Source: §I · p. 2
• NOMA superposes users’ messages, orders users by effective channel gains, and allocates a higher fraction of power to weaker users.Source: §I · p. 2
• 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.Source: §I · pp. 2–3

Prior work

• Prior work studied NOMA mainly in SISO or limited MIMO settings, including outage, fairness, SIC error propagation, and fixed power allocation.Source: A. Literature · p. 3

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.Source: B. Contributions · p. 3

Contributions

• The proposed MMA iteratively solves an SOCP with decodability and non-zero-rate constraints, replacing traditional SDP-based approaches.Source: B. Contributions · p. 4
• The algorithm is reported to converge in few iterations, has polynomial worst-case complexity, and under plausible assumptions converges to a KKT point.Source: B. Contributions · p. 4

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.Source: B. Contributions · p. 4

Organization

• The paper proceeds from system modeling and preliminaries to algorithm development, reduced-complexity approximation, numerical results, and conclusions.Source: C. Structure · p. 5

System model

• The downlink system has a BS with T antennas serving N single-antenna users under NOMA transmission.Source: §II · p. 5

Signal model

• Each user’s symbol is multiplied by a complex precoding vector, and the BS transmits the superposition of all users’ weighted messages.Source: §II · p. 5
• Under frequency-flat channels, UE-i receives its channel-weighted superposition signal plus complex Gaussian noise.Source: §II · p. 5

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.Source: §II · p. 6, Figure 1 · p. 6

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.Source: §II · p. 6

• UE-1 is assumed to be the weakest user and therefore cannot decode interfering signals.Source: A. Problem Formulation · p. 7
• Users are ordered by increasing index, although ordering by channel strength may not be optimal and alternative orders may achieve better rates.Source: A. Problem Formulation · p. 7
• Higher-ranked users must satisfy decoding-rate constraints for messages from weaker users to enable successive interference cancellation.Source: A. Problem Formulation · p. 7
• The formulation maximizes sum rate while enforcing ordered received-signal strengths and a total transmit-power bound Pth.Source: A. Problem Formulation · p. 8

• The original optimization problem is transformed before applying iterative approximations to improve tractability.Source: §III · p. 8
• The transformed problem remains intractable because several constraints are non-convex, although the remaining constraints are convex and admit SOC representation.Source: A. Equivalent Transformations · pp. 8–9
• The reformulation introduces auxiliary variables for rates and decoding constraints while preserving equivalent relationships among SINR terms.Source: A. Equivalent Transformations · pp. 8–9
• The geometric mean of the rate vector is concave and increasing, and can be represented using second-order cone constraints.Source: A. Equivalent Transformations · p. 9
• The non-convex constraints are handled separately after factoring the formulation into distinct constraint groups.Source: A. Equivalent Transformations · p. 9

• The convex-concave procedure, also called minorization-maximization, is used to handle the non-convex constraints.Source: B. Approximation of the non-convex… · p. 11
• The same approximation strategy is applied successively to the non-convex constraint sets in (14b), (14c), (14d), and (14e).Source: B. Approximation of the non-convex… · pp. 11–13
• Bilinear and quadratic terms are replaced by first-order Taylor approximations around values from the current iteration.Source: B. Approximation of the non-convex… · pp. 11–12
• The approximation preserves a minorant-based iterative structure in which the current point defines the next surrogate iterate.Source: B. Approximation of the non-convex… · p. 12
• The resulting constraint involving the approximated terms is convex in the variables of interest.Source: B. Approximation of the non-convex… · p. 12

The Proposed Solution

• Each MMA iteration solves an optimization problem constrained by the current convexified formulations and auxiliary-variable definitions.Source: §IV · p. 13, A. The Procedure · p. 13
• The algorithm’s sequence of iterates remains feasible for the original problem, with each iterate belonging to the initial feasible set.Source: 1) Convergence · p. 14

Convergence

• The objective values are non-decreasing, satisfying O_t+1 ≥ O_t, and therefore converge.Source: 1) Convergence · p. 14
• Under convexity and compactness of the feasible set, the algorithm converges to a finite value.Source: 1) Convergence · p. 15
• As t tends to infinity, the algorithm converges to a KKT point of the original formulation (14).Source: 2) KKT Conditions · p. 15

Reduced Complexity Approximation

• The reduced formulation omits N(N −1)/2 SINR terms and correspondingly removes N^2 −N inequality constraints.Source: §V · p. 15
• When channels are clearly ordered and the stated channel-ratio inequalities hold, the reduced and original formulations are approximately equivalent.Source: §V · pp. 16–17
• The probability that the stronger-user SINR exceeds the weaker-user SINR increases as the channel-strength ratio becomes more favorable.Source: §V · p. 17

• Each MMA iteration solves an SOCP, and the algorithm’s worst-case complexity is determined by the SOCP solved at each step.Source: A. Complexity · p. 18
• General interior-point SOCP complexity depends on the numbers of constraints and variables and the dimensions of SOC constraints.Source: A. Complexity · p. 18
• 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.Source: A. Complexity · p. 18
• The SOCP complexity estimates are plotted as functions of both N and T, showing how complexity increases with system dimensions.Source: A. Complexity · p. 19
• C-NOMA uses the SOCP in (29), whereas A-NOMA uses the SOCP in (35).Source: A. Complexity · p. 19

• 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.Source: §VI · pp. 19–20
• 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.Source: §VI · p. 20
• ZF performs poorly at lower SNRs but improves at sufficiently high transmit power as distance-induced channel ill-conditioning is partially circumvented.Source: §VI · pp. 20–21
• For N = T = 4, A-NOMA and C-NOMA overlap at low SNRs, whereas C-NOMA outperforms A-NOMA at higher transmit SNR.Source: §VI · pp. 21–22
• 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.Source: §VI · p. 22
• Reducing D0 from 50 m to 10 m considerably enhances ZF sum rates by minimizing path loss and improving channel conditioning.Source: §VI · p. 23
• When N > T = 3, randomly selecting three users can make C-NOMA underperform because effective multiuser diversity is likely lost.Source: §VI · p. 24

• The study addresses sum rate maximization in a MISO downlink system using NOMA.Source: §VII · p. 25
• The non-convex optimization problem is approximated with a minorization-maximization method, solving an SOCP at each iteration.Each step has polynomial computational complexity.Source: §VII · p. 25
• The proposed algorithm numerically converges within a few iterations in the considered scenarios.Source: §VII · p. 25
• A reduced-complexity approximation is developed, together with conditions under which it is tight and an analysis of its tightness.Source: §VII · p. 25
• NOMA has superior performance compared to conventional orthogonal multiple access schemes, obtaining high data rates with small transmit power.Source: §VII · p. 25
• 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.Source: §VII · p. 25

• The surrogate functions replacing non-convex terms are non-decreasing across iterations, yielding nested feasible sets with F_t+1 ⊇ F_t.Source: Appendix B · p. 26
• Because the feasible sets expand monotonically, the objective sequence is non-decreasing and may converge to a positive value.Source: Appendix B · p. 26
• The analysis assumes that the variable sequence generated by the algorithm converges to a value V∗.Source: Appendix C · p. 26
• It also assumes that the constraints in the approximate problem are qualified at the limiting point.Source: Appendix C · p. 26
• The appendix represents convex constraints abstractly and obtains approximated constraints by replacing non-convex functions with convex minorants.Source: Appendix C · p. 26
• At convergence, the KKT conditions of the approximate problem reduce to those of the original problem and the simplified problem.Source: Appendix C · pp. 26–27

• For arbitrary user and node indices, the derivation uses assumptions on the noise variances and recursively related channel coefficients.Source: Appendix D · p. 27
• With random unitary precoding, unitary transformations preserve the complex Gaussian distribution of the channel variables.Source: Appendix E · p. 27
• The resulting x_i^k variable is exponentially distributed, while y_i^k follows a Chi-square distribution.Source: Appendix E · p. 28
• The appendix derives the cumulative distribution function and probability density function of SINR_i^k from these component distributions.Source: Appendix E · p. 28
• The probability calculation is completed using standard referenced distribution identities after applying the unitary Gaussian transformation.Source: Appendix E · p. 28