This paper focuses on radar waveform optimization for minimizing the Cramér-Rao bound (CRB) in a multiple-input multiple-output (MIMO) radar system. In contrast to conventional approaches relying on semi-definite programming (SDP) and optimization toolboxes like CVX, we introduce a pioneering and efficient waveform optimization approach in this paper. Our proposed algorithm first applies sequential linear approximation to transform the original CRB-based problem with the transmit power constraint into a sequence of convex subproblems. By introducing a proximal term and further leveraging the Karush-Kuhn-Tucker (KKT) conditions, we derive the optimal closed-form solution for each subproblem. The convergence of the proposed algorithm is then proved rigorously. Numerical results demonstrate that the proposed approach significantly reduces computational complexity -- at least two orders of magnitude lower than the baseline algorithms while maintaining the same radar sensing accuracy.