A novel selection algorithm for maximizing genetic response while constraining the rate of inbreeding is presented. It is shown that the proposed method controls the rate of inbreeding by maintaining the sum of squared genetic contributions at a constant value and represents an improvement on previous procedures. To maintain a constant rate of inbreeding the contributions from all generations are weighted equally and this is facilitated by modifying the numerator relationship matrix. By considering the optimization of the contributions of many generations the initial mating proportions (the genetic contributions to the next generation) are not equal to their long-term values, but are set equal to the expected long-term contributions given the current information. This is confirmed by the regression of the long-term contributions on the assigned mating proportions being close to one. The gain obtained from the selection algorithm is compared with the maximum theoretical genetic gain under constrained inbreeding. It is concluded that this theoretical upper bound is in general unattainable, but from this a concept of genetic efficiency in terms of resources and constraints is derived.