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Visible light communication (VLC) is a promising technology for the next generation wireless communication systems due to its high spectral efficiency and energy efficiency. In this article, a secure multiple-input single-output (MISO) VLC network is studied, where simultaneous lightwave information and power transfer (SLIPT) is exploited to support multiple energy-limited devices tacking into account a practical non-linear energy harvesting model. The transmit power minimization and the minimum secrecy rate maximization problems are formulated under both perfect and imperfect channel state information, respectively. To further improve the user connectivity, those problems are also investigated in MISO-VLC networks with non-orthogonal multiple access (NOMA). To tackle these challenging non-convex problems, semidefinite program relaxation and <inline-formula> <tex-math notation="LaTeX">\mathcal {S} </tex-math></inline-formula>-Procedure are exploited. It is proved that optimal beamforming schemes can be obtained for the considered two types problems in MISO-VLC SLIPT networks, while a sub-optimal solution can be obtained for the transmit power minimization problem in those networks with NOMA. It is found that there is a non-trivial trade-off between the average harvested power and maximum-minimum secrecy rate. Moreover, it is shown that the performance achieved with NOMA outperforms that of conventional orthogonal multiple access in MISO-VLC SLIPT networks, despite the existence of imperfect channel state information.