1 Max-Min Rate Optimization for Uplink IRS-NOMA with Receive Beamforming Luiggi Cantos, Muhammad Awais, and Yun Hee Kim, Senior Member, IEEE Abstract—This paper 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 optimiza- tion. 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. Index Terms—Intelligent reflecting surface, nonorthogonal multiple access, power allocation, receive beamforming I. I NTRODUCTION For 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-of-sight (LoS) chan- nels 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]–[14]. For the plat- form, a multiantenna base station (BS) is indispensable so that its beamforming (BF) has been optimized jointly with IRS reflection [3], [4], [6]–[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. L. Cantos, M. Awais, and Y.H. Kim are with the Department of Electronics and Information Convergence Engineering, Kyung Hee University, Yongin 17104, Korea (e-mail: {lrcantos, mawais, yheekim}@khu.ac.kr). This research was supported by the National Research Foundation of Korea (NRF) under Grant NRF-2021R1A2C1005869 and by the Institute of Information & Communications Technology Planning & Evaluation (IITP) under the Information Technology Research Center (ITRC) support program IITP-2021-0-02046, with funding from the Ministry of Science and ICT (MSIT), Korea (Corresponding author: Yun Hee Kim). For the uplink IRS-NOMA, IRS reflection was primarily optimized with a single-antenna BS [10]–[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 although 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 prob- lems 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 approximat- ing 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 effi- cient 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 This article has been accepted for publication in IEEE Wireless Communications Letters. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/LWC.2022.3206903 © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: Kyunghee Univ. Downloaded on November 25,2022 at 09:36:49 UTC from IEEE Xplore. Restrictions apply.