Magnetic random-access memory (MRAM) is a promising memory technology due to its high density, non-volatility, and high endurance. However, achieving high memory fidelity incurs significant write-energy costs, which should be reduced for large-scale deployment of MRAMs. In this paper, we formulate a \emph{biconvex} optimization problem to optimize write fidelity given energy and latency constraints. The basic idea is to allocate non-uniform write pulses depending on the importance of each bit position. The fidelity measure we consider is mean squared error (MSE), for which we optimize write pulses via \emph{alternating convex search (ACS)}. By using Karush-Kuhn-Tucker (KKT) conditions, we derive analytic solutions and propose an \emph{iterative water-filling-type} algorithm by leveraging the analytic solutions. Hence, the proposed iterative water-filling algorithm is computationally more efficient than the original ACS while their solutions are identical. Although the original ACS and the proposed iterative water-filling algorithm do not guarantee global optimality, the MSEs obtained by the proposed algorithm are comparable to the MSEs by complicated global nonlinear programming solvers. Furthermore, we prove that the proposed algorithm can reduce the MSE exponentially with the number of bits per word. For an 8-bit accessed word, the proposed algorithm reduces the MSE by a factor of 21. We also evaluate the proposed algorithm for MNIST dataset classification supposing that the model parameters of deep neural networks are stored in MRAMs. The numerical results show that the optimized write pulses can achieve \SI{40}{\%} write energy reduction for a given classification accuracy.