This paper develops a novel optimization of partial transmit sequences (PTS) with phase quantization to reduce the peak-to-average-power ratio (PAPR) of orthogonal frequency-division multiplexing (OFDM) signals and enable data detection without side information. Since the formulated optimization problem is non-convex and has exponential time complexity, we employ a convex relaxation method to convert the original optimization problem into a series of convex programming whose solutions converge to a sub-optimal point satisfying the Karush-Kuhn-Tucker (KKT) conditions in polynomial time. Moreover, the obtained phase factors are quantized to enable the maximum likelihood (ML) phase estimation at the receiver, hence removing the need of sending side information. Analytical and numerical results are provided to show that our proposed PTS design achieves better PAPR reduction over existing PTS methods, while no performance degradation is incurred in data detection when the number of phase factors is properly chosen.