The capability of Multiple-Input Multiple-Output (MIMO) radar systems to adapt waveforms across antennas allows flexibility in transmit beampattern design. In cognitive radar, a popular cost function is to minimize deviation against an idealized beampattern (which is arrived at with knowledge of the environment). The optimization of transmit beampattern becomes particularly challenging in the presence of practical constraints on the transmit waveform. In this work, we study two important but difficult non-convex constraints: constant modulus and orthogonality of waveform code across transmit antennas. Individually, incorporating these constraints continues to be challenging and they have rarely been jointly studied because each set is individually a (non-convex) manifold. We develop a new approach by leveraging the theory of optimization over manifolds: we derive a new projection, descent and retraction (PDR) update strategy that allows for monotonic cost function improvement while maintaining feasibility over the complex circle manifold (constant modulus set). The approach allows a tractable extension to incorporate orthogonality as a penalty term with the cost function. We provide analytical guarantees of monotonic cost function improvement along with proof of convergence. Experimentally, we show that PDR can outperform state of the art wideband beampattern design methods while being computationally tractable. Robustness to target direction mismatch (enabled by orthogonal waveforms) is also demonstrated.