Max-Min Rate Optimization for Uplink IRS-NOMA With Receive Beamforming

This letter addresses an intelligent reflecting surface (IRS) to the uplink nonorthogonal multiple access (NOMA) served by a multiantenna receiver for effective data collection from massive devices. We aim to achieve max-min fairness of the network by optimizing receive beamforming, IRS reflection, and transmit power allocation (PA) of the devices. For this purpose, first, we design a block coordinate descent (BCD) algorithm that reduces the complexity of a conventional IRS reflection optimization. Next, we design a nonlinear optimization (NLO) problem solvable with the limited-memory Broyden-Fletcher-Goldfarb-Shanno bounded (L-BFGS-B) algorithm, which is renowned for handling large-scale problems, to cope with large IRS elements and devices. The problem is formed with a smooth but complex objective function that depends on the IRS phase shift and PA vectors for which the gradient is derived in a computationally efficient form. The results reveal that the proposed BCD and proposed NLO with the L-BFGS-B outperform the conventional BCD in performance and complexity, where the NLO approach offers a substantial complexity reduction.


I. INTRODUCTION
F OR 6G networks, intelligent reflecting surfaces (IRSs) have emerged as one of the most potential technologies due to their capability of tailoring channel environments favorable to the desired performance metric [1]. An IRS comprising passive elements can be cost-effectively deployed while providing seamless communication by creating line-ofsight (LoS) channels when direct channels are unavailable due to blockages. These advantages have led to the application of IRSs to various multiuser communications, for which their achievable performance and essential implementation approaches, such as IRS reflection optimization, have been investigated [2].
Particularly IRS-assisted nonorthogonal multiple access (IRS-NOMA) has been extensively investigated to provide an effective platform that allows massive access of explosively increasing Internet-of-Things devices [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. For the platform, a multiantenna base station (BS) is indispensable so that its beamforming (BF) has been optimized jointly with IRS reflection [3], [4], [6], [7], [8], [13], [14]. For the downlink IRS-NOMA, to leverage a performance metric such as sum rate [3], energy efficiency [4], reduction of power consumption [6], [8], and minimum rate [7], IRS reflection was jointly optimized with transmit BF for a predetermined successive interference cancellation (SIC) order. For the uplink IRS-NOMA, IRS reflection was primarily optimized with a single-antenna BS [10], [11], [12] and later with a multiantenna BS [13], [14] to maximize the sum rate. These studies except for [10] did not consider power allocation (PA) of the devices and SIC order since the sum rate without any rate constraint is maximized with the maximum PA irrelevant to SIC order. By introducing the rate constraints making the sum rate dependent on PA and SIC order, the sum rate maximization problem was solved by optimizing the IRS reflection and PA through block coordinate descent (BCD), also called alternating optimization (AO), for a predetermined SIC order [10]. However, a max-min fairness problem with a rather complex objective function depending on PA and SIC order has not been investigated for the uplink IRS-NOMA yet al.hough it was tackled for the downlink case where the transmit BF and IRS reflection were optimized through BCD for a predetermined SIC order [7].
Thus, for the uplink IRS-NOMA served by a multiantenna BS, we consider a max-min rate fairness problem. We optimize the receive BF, IRS reflection, and PA of the devices for a given SIC order, where the SIC order is determined initially as in [7] and can be updated for further optimization. The primary contributions are summarized as follows: • First, we develop the BCD algorithm comprising a closed-form solution for the receive BF, a successive convex approximation (SCA) method for IRS reflection, and a bisection search employing linear feasibility problems for the PA. This algorithm differs from that in [7] optimizing IRS reflection and transmit BF resorting to the bisection search with semidefinite relaxation (SDR), and that in [10] optimizing IRS reflection and PA using SDR and linear program, respectively. The proposed BCD with SCA for IRS reflection optimization is demonstrated to outperform the BCD with the SDR-based bisection search. • Next, we propose a reformulated problem to fit into a nonlinear optimization (NLO) solver for many IRS elements and devices. The problem is formulated with the IRS phase shift and PA vectors by incorporating the closed-form receive BF into the rates and approximating the minimum in the objective to the LogSumExp (LSE) function employed to approximate the sum of the multicast rates in [15]. The resultant problem with a smooth objective and bounded constraints can be solved effectively with the limited-memory Broyden-Fletcher-Goldfarb-Shanno bounded (L-BFGS-B) algorithm [16], [17] renowned for a large-scale problem. • For faster optimization, we derive the gradient of the complicated objective function in a computationally efficient form for the L-BFGS-B and obtain the initial SIC order by using the L-BFGS-B instead of SDR [7]. The proposed NLO with the L-BFGS-B significantly reduces the computational complexity of the BCD algorithms, enabling a repeated optimization with the updated SIC order for further enhancement. Notation: C n×m (R n×m ) denotes the sets of n × m complex-valued (real-valued) matrices with C n×1 = C n (R n×1 = R n ). We use diag(s) for the diagonal matrix with a diagonal vector s and [s] n for the nth entry of a vector s; tr(C), [C] n,m , and [C] :,m denote the trace, (n, m)th element, and mth column of a matrix C, respectively. We use E[ · ] for expectation, CN (μ, Σ) for complex Gaussian with mean μ and covariance matrix Σ, and U (a, b) for a uniform distribution over (a, b). Fig. 1, where a BS equipped with M antennas collects data from K single-antenna devices with the help of an IRS. Here, M can take any natural number although M < K is generally assumed for the uplink NOMA. The IRS consists of N passive elements with reflection coefficients θ = [θ 1 θ 2 · · · θ N ] T ∈ C N . We assume unit modulus reflection as |θ n | = 1, or equivalently θ n = e j φn for φ n ∈ [0, 2π) and n ∈ N {1, 2, . . . , N } which will be expressed as θ = e j φ with the phase shift vector

II. SYSTEM MODEL AND PROBLEM FORMULATION Consider an uplink IRS-NOMA demonstrated in
Each device k transmits its symbol x k at power p k simultaneously for k ∈ K, which is received at the BS as follows.
where n ∼ CN (0, σ 2 I M ) is the background noise and is the equivalent channel from device k to the BS with A k = Gdiag(f k ). The BS detects the symbols from (1) with receive BF matrix W ∈ C M ×K accompanied by SIC. Let π k in π = [π 1 , π 2 , . . . , π K ] T represent the SIC order of device k. The signal for detecting x k after SIC is expressed as with the receive BF vector w k = [W] :,k for device k. Thus, the signal-to-interference-plus-noise ratio (SINR) is given by resulting in the achievable rate as R k = log 2 (1 + γ k ). We aim to maximize the minimum rate of the devices for fairness by optimizing the receive BF W, IRS reflection θ (or φ), PA p = [p 1 , p 2 , . . . , p K ] T , and SIC order π as follows: where Π represents the set of all possible permutations of π. We predetermine the SIC order π based on the channel power is not yet determined, we initially obtain the maximally achievable channel power Ξ † k by solving max and construct π in descending order of Ξ † k . We optimize (W, θ , p) through an algorithms presented in the following sections using this initial π . If necessary, after obtaining the solution (θ , p ), we calculate the actual channel power Ξ k (p k , θ ) to update the SIC order with which (W, θ , p) can be optimized again if the SIC order has been changed.
In the following, we present two different methods for finding a solution of (W, θ , p) for a given SIC order.

III. BLOCK COORDINATE DESCENT (BCD) ALGORITHM
This section presents the BCD algorithm that optimizes W, θ, and p iteratively. For a given SIC order, we rewrite problem (5) by replacing where the arguments (w k , θ , p) of γ k are added for clarity.
First, we obtain the optimal receive BF W by solving (7) for given (θ , p), which is equivalent to solve max w k ∈C N γ k (w k , θ , p) for each k. The optimal receive BF is given by [18], [19] w Next, we optimize the IRS reflection θ for given (W, p) as which is transformed into max using (4) andσ 2 k = σ 2 w k 2 . Problem (11) can be solved through the bisection search resorting to the SDR employing X = [θ T 1] T [θ H 1] and Gaussian randomization as in [7]. To reduce the high complexity incurred by the SDR, we instead find a way of applying SCA by rewriting (11b) as where μ k (θ ) Then, the nonconvex term μ k (θ ) is approximated by its first-order Taylor expansion aroundθ + aŝ ] is the Wirtinger derivative and {·} represents the real part. By relaxing (11c) further, we finally obtain the SCA problem at the ith iteration using the previous solutionθ †  (14) is then projected to a feasible one as The SCA is repeated until convergence or the maximum number I θ of iterations is reached to output the final solution θ † * . Finally, we formulate the PA problem for given (W, θ ) as max with μ l,k = |w H k h l (θ )| 2 , which is a linear program for a given t p . By solving a sequence of linear feasibility problems as in [19], the optimal solution (p † , t † p ) is found. Algorithm 1 summarizes The proposed BCD algorithm for a given SIC order. The SCA in the BCD is calculated at complexity O(N 3.5 I θ ) by solving SOCP [15], while the SDR requires O(N 4.5 I X ) with I X iterations for the SDR-based bisection search [7] when N K. Therefore, the complexity of the proposed SCA-based BCD can be approximated as O(((M 3 + MNK ) + N 3.5 I θ + K 3.5 I p )I bcd ), where I p represents the number of bisection search iterations for PA.

IV. NONLINEAR OPTIMIZATION WITH L-BFGS-B
This section provides a computationally efficient solution for large N and K by reformulating the problem with a smooth objective function and simple bounded constraints for NLO.

6:
Compute W (l) with (8) for (θ , p) = (θ (l) , p (l) ). 7: until the maximum number I bcd of iterations is reached or the objective value of (7) is converged. 8: Output: θ = θ (l) , p = p (l) , and W = W (l) . respect to (φ, p). Then, we reformulate problem (5) for a given SIC order without receive BF as max , ∀k (17b) which has only simple bounded constraints. Next, we approximate the minimum to the LSE function as [15] f where the maximum error ln K ρ is set to 10 −3 with ρ = 10 3 ln K in the following. Therefore, we propose an approximate problem for (17) as max to be tackled by the L-BFGS-B algorithm, which is renowned for a large-scale problem having a differentiable objective under bounded constraints. The L-BFGS-B is a quasi-Newton approach that updates a new search point with the gradient and an approximate inverse Hessian estimated with m B recent search points and their gradients stored [16], [17]. Therefore, the gradient of the objective with respect to , p)] T , is a crucial driver in solving (19) with the L-BFGS-B. Then, we derive the gradient in a computationally efficient form for efficient implementation. For where c(φ, p) = 1/[ ln 2 K k =1 (1 + Γ k (φ, p)) −ρ ]. Then, the gradient is expressed by a matrix multiplication as The proposed NLO runs the L-BFGS-B algorithm using the derived gradient ∇ x f (φ, p), starting from a randomly chosen initial point in the feasible set of (φ, p) until the maximum number I nlo of iterations is reached or the objective is converged. Note that problem (6) for the initial SIC order can be solved by the L-BFGS-B using the gradient for δ ≈ 0 and the standard unit vector e i with [e i ] i = 1 is calculated at O((N +K )M 3 K ) due to the matrix inversion for each entry. Thus, the L-BFGS-B with the derived gradient is more effective, leading to the total complexity O((M 3 K +MNK +K 2 +m B (N +K ))I nlo ) after incorporating O(m B (N + K )) for a search point update.

V. SIMULATION RESULTS
The performance is assessed in a setup similar to that in [3], [4], [10]; the BS and IRS are positioned at (0, 0, 10) and (50, 20, 10), respectively, in the (x, y, z) coordinates in meters while the devices are located randomly at (x, y, 1.  [7]. For performance evaluation, we generate 1000 channel samples with P max k = 20 dBm, σ 2 = −100 dBm, and 1 MHz bandwidth [3], [10] at an operating frequency of 2.5 GHz [7]. The algorithms in Sections III and IV are called BCD-SCA (BCD-SDR if the SDR-based bisection search is used instead of SCA) and L-BFGS-B, respectively, where the SIC order is predetermined with (6) unless stated otherwise. We set I θ = I p = 30 and I bcd = 23 for BCD-SCA (I X = 30 and 100 Gaussian samples for BCD-SDR) while I nlo = 5000 and m B = 7 for L-BFGS-B, at an error tolerance of = 10 −6 .  Table I also compares the average computational time per sample spent on Intel i9-12900K @3.19 GHz CPU. In Fig. 2, the performance of L-BFGS-B (IRS) optimizing the IRS phase vector at maximum PA and No-IRS (PA) optimizing PA without any IRS is also shown for the benchmarks. While the BCD algorithms are shown up to N = 64 due to their unacceptable complexity for a large N, the time-efficient NLO using L-BFGS-B is combined with more complex SIC ordering approaches; BestOrd finds the best SIC order by performing K ! optimization for all possible SIC orders and IterOrd updates the SIC order with the optimized values for a new optimization. The results show that the predetermined SIC order based on (6) provides a performance close to that of the optimal one for a large N, which renders the update of the SIC order unnecessary when K = 4. L-BFGS-B offers the best performance with the predetermined SIC order, followed by BCD-SCA and  Table I. It is also observed that IRS reflection optimization dominates the performance for a large N, making PA unnecessary.
The average max-min rate is compared as the number of antennas and devices increases as M = K/2 in Figs. 3(a)-3(c) for N = 32, 64, and 128, respectively; L-BFGS-B/BestOrd is not shown due to the high complexity in K ! optimizations. The L-BFGS-B performs the best with the gain increasing with M (= K/2) over the BCD algorithms. Furthermore, a slight performance enhancement is observed by updating the SIC order for L-BFGS-B (L-BFGS-B/IterOrd) as K increases. PA dominates the performance more than IRS for a larger K when L-BFGS-B (IRS) and No-IRS (PA) are compared.
Finally, the performance of L-BFGS-B is demonstrated as a function of the number K of devices while fixing M = 4 and varying N from 64 to 512 in Fig. 4. At a fixed number of antennas, as K increases, the average max-min rate, E [R min ], in the left subfigure decreases; however, the total max-min rate supported by the network, E [K R min ], in the right subfigure increases by serving more devices at the same resource. The results reveal that the total max-min rate of the network can be increased by about 24 % -40 % by increasing N from 64 to 512 in serving up to K = 32 devices.

VI. CONCLUSION
We considered a max-min rate fairness problem for the uplink IRS-NOMA, which optimizes receive BF, IRS reflection, and PA of devices jointly for the predetermined SIC order. First, we devised a BCD algorithm optimizing three sub-blocks iteratively through the closed-form receive BF, SCA-based IRS reflection optimization, and PA solved with linear feasibility problems. Next, we proposed a reformulated problem with a smooth but complicated objective function of the IRS phase shift and PA vectors under bounded constraints for fast optimization. The problem was solved with the L-BFGS-B algorithm accompanied by a gradient derived in a closed form. The results revealed that the devised algorithms outperform the benchmark, where the L-BFGS-B offers a better solution at a significantly lower computational time and thus enables the SIC order updating for possible performance gain. Future research should focus on the impact of the imperfect SIC on this optimization framework and performance.

APPENDIX
We express the SINR for the Jacobian explicitly as from Γ k (φ, p) = γ k (w † k , e j φ , p). In following derivation, the arguments of h k and B k will be omitted. We first obtain Finally, (22) is obtained by replacing (26) and (27) in (25). We next consider ∂Γ k ∂p i for i ∈ K. Since B k (θ , p) in (9) depends only on {p l } π l >π k , we have ∂Γ k ∂p i = 0 for π i < π k and ∂Γ k ∂p i = h H k B −1 k h k for π i = π k . For π i > π k ,