{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T05:24:26Z","timestamp":1787117066398,"version":"build-2736575974"},"reference-count":55,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T00:00:00Z","timestamp":1786924800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Jiangsu Province of China","award":["BK20240593"],"award-info":[{"award-number":["BK20240593"]}]},{"name":"Natural Science Foundation of the Higher Education Institutions of Jiangsu Province","award":["24KJB470018"],"award-info":[{"award-number":["24KJB470018"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Joint active and passive beamforming in an intelligent reflecting surface (IRS)-assisted non-orthogonal multiple-access (NOMA) downlink is a non-convex constrained problem whose Karush\u2013Kuhn\u2013Tucker (KKT) points are generally non-isolated. We formulate the normalized beamformers, continuous IRS angles, and inequality multipliers as one coupled projected primal\u2013dual dynamical state. Here, single-layer means simultaneous primal\u2013dual evolution; the numerical integrator may still contain multiple stages. Angle coordinates remove the unit-modulus equalities, while an adjacent-chain representation reduces the received-power ordering constraints without changing their feasible set for a fixed decoding order. We prove equilibrium\u2013KKT equivalence and local transverse exponential attraction of the KKT equilibrium set under strict complementarity, locally constant active rank, nonzero streams, a quotient second-order condition, and a sufficiently large finite penalty parameter. A stiff-ODE warm-up followed by a nonmonotone semismooth pseudo-transient continuation (PTC) method computes a steady-state root.<\/jats:p>","DOI":"10.3390\/s26165211","type":"journal-article","created":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T16:35:08Z","timestamp":1786984508000},"page":"5211","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["JODS-PD: Single-Layer Primal\u2013Dual Dynamical Optimization for IRS-Assisted NOMA Beamforming"],"prefix":"10.3390","volume":"26","author":[{"given":"Yitong","family":"Shi","sequence":"first","affiliation":[{"name":"School of Communication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210023, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pei","family":"Peng","sequence":"additional","affiliation":[{"name":"School of Communication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210023, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhiyuan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210023, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5394","DOI":"10.1109\/TWC.2019.2936025","article-title":"Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming","volume":"18","author":"Wu","year":"2019","journal-title":"IEEE Trans. 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