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Estimation of the channel state information (CSI) in quadratic form (i.e., quadratic channel estimation) in the downlink can be performed at the base station by using the relayed signals from the mobile users, which facilitates optimization with transmitter CSI. In this letter, the condition for the optimal training sequence for quadratic channel estimation in a multiuser multiple-input single-output (MISO) antenna system in the downlink is first obtained. The mean-square-error (MSE) in the CSI estimate is then analyzed. Based on the quadratic CSI estimates, a robust beamforming optimization algorithm to minimize the base station power while achieving individual users' quality-of-service (QoS) constraints, measured by the MSE in data reception, is proposed.