Controlling inbreeding and maximizing genetic gain using semi-definite programming with pedigree-based and genomic relationships

Because of the relatively high levels of genetic relationships among potential bull sires and bull dams, innovative selection tools should consider both genetic gain and genetic relationships in a long-term perspective. Optimum genetic contribution theory using official estimated breeding values for a moderately heritable trait (production index, Index-PROD), and a lowly heritable functional trait (index for somatic cell score, Index-SCS) was applied to find optimal allocations of bull dams and bull sires. In contrast to previous practical applications using optimizations based on Lagrange multipliers, we focused on semi-definite programming (SDP). The SDP methodology was combined with either pedigree (aij) or genomic relationships (fij) among selection candidates. Selection candidates were 484 genotyped bulls, and 499 preselected genotyped bull dams completing a central test on station. In different scenarios separately for PROD and SCS, constraints on the average pedigree relationships among future progeny were varied from aij = 0.08 to aij = 0.20 in increments of 0.01. Corresponding constraints for single nucleotide polymorphism-based kinship coefficients were derived from regression analysis. Applying the coefficient of 0.52 with an intercept of 0.14 estimated for the regression pedigree relationship on genomic relationship, the corresponding range to alter genomic relationships varied from fij = 0.18 to fij = 0.24. Despite differences for some bulls in genomic and pedigree relationships, the same trends were observed for constraints on pedigree and corresponding genomic relationships regarding results in genetic gain and achieved coefficients of relationships. Generally, allowing higher values for relationships resulted in an increase of genetic gain for Index-PROD and Index-SCS and in a reduction in the number of selected sires. Interestingly, more sires were selected for all scenarios when restricting genomic relationships compared with restricting pedigree relationships. For example, at constraint of fij = 0.185 and selection on Index-PROD, the number of selected sires was 35. In contrast, only 21 sires were selected at the comparable constraint on additive genetic relationship of aij = 0.09. A further reduction in relationships is possible when using SDP output (i.e., suggested genetic contributions of selected parents) and applying a simulated annealing algorithm to define specific mating plans. However, the advantage of this strategy is limited to a short-term perspective and probably not successful in the period of genomic selection allowing a substantial reduction of generation intervals.