Evaluation criterion for response to selection with constraint

The aims of the present study are to represent the concept of restricted breeding values algebraically and to propose a criterion for evaluating the genetic responses achieved by using a restricted selection procedure. An additive genetic mixed model characterized by multiple traits with constraints was assumed. If the random errors approach zero and the fixed effects can be completely estimated correctly in the model, the restricted best linear unbiased predictor of breeding values (u _R ) is equal to [I _q -G ₀ C ₀ (C ^' ₀ G ₀ C ₀ ^-1 )C ^' ₀ I], where G ₀ , C ₀ , and u are the additive genetic variance-covariance matrix for the q traits, the matrix for restriction, and the vector of breeding values, respectively. Therefore, if we want to evaluate the response to restricted selection, such as by a stochastic computer simulation study with known breeding values, we can use u _R as only one criterion.