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In the past, ad hoc methods have been used to choose gains in proportionate-type normalized least mean-square algorithms without strong theoretical under-pinnings. In this correspondence, a theoretical framework and motivation for adaptively choosing gains is presented, such that the mean-square error will be minimized at any given time. As a result of this approach, a new optimal proportionate-type normalized least mean-square algorithm is introduced. A computationally simplified version of the theoretical optimal algorithm is derived as well. Both of these algorithms require knowledge of the mean-square weight deviations. Feasible implementations, which estimate the mean-square weight deviations, are presented. The performance of these new feasible algorithms are compared to the performance of standard adaptive algorithms when operating with sparse, non-sparse, and time-varying impulse responses, when the input signal is white. Specifically, we consider the transient and steady-state mean-square errors as well as the overall computational complexity of each algorithm.