There is a fundamental trade-off between the channel representation resolution of codebooks and the overheads of feedback communications in the fifth generation new radio (5G NR) frequency division duplex (FDD) massive multiple-input and multiple-output (MIMO) systems. In particular, two types of codebooks (namely Type I and Type II codebooks) are introduced with different resolution and overhead requirements. Although the Type I codebook based scheme requires lower feedback overhead, its channel state information (CSI) reconstruction and beamforming performance are not as good as those from the Type II codebook based scheme. However, since the Type I codebook based scheme has been widely used in 4G systems for many years, replacing it by the Type II codebook based scheme overnight is too costly to be an option. Therefore, in this paper, using Type I codebook, we leverage advances in cutting plane method to optimize the CSI reconstruction at the base station (BS), in order to close the gap between these two codebook based beamforming schemes. Numerical results based on channel samples from QUAsi Deterministic RadIo channel GenerAtor (QuaDRiGa) are presented to show the excellent performance of the proposed algorithm in terms of beamforming vector acquisition.