Multiplicative algorithms are simple optimization schemes that iteratively adjust the coordinates of an input probability vector by multiplicative factors so as to increase a suitable objective function.  Examples include the EM algorithm for maximum likelihood estimation of mixture proportions, multiplicative algorithms for computing approximate optimal designs, and the Arimoto-Blahut algorithm for calculating channel capacities in Shannon theory.  By exploiting the connections between these seemingly separate problems we derive general conditions that ensure monotonic convergence for multiplicative algorithms in optimal designs, and construct hybrid algorithms that converge faster but maintain the simplicity and stability.