  An important problem in tree breeding is optimal selection from candidate
pedigree members to produce the highest performance in seed orchards, while
conserving essential genetic diversity. The most beneficial members should
contribute as much as possible, but such selection of orchard parents would
reduce performance of the orchard progeny due to serious inbreeding. To avoid
inbreeding, we should include a constraint on the numerator relationship matrix
to keep a group coancestry under an appropriate threshold. Though an SDP
(semidefinite programming) approach proposed by Pong-Wong and Woolliams gave an
accurate optimal value, it required rather long computation time.
  In this paper, we propose an SOCP (second-order cone programming) approach to
reduce this computation time. We demonstrate that the same solution is attained
by the SOCP formulation, but requires much less time. Since a simple SOCP
formulation is not much more efficient compared to the SDP approach, we exploit
a sparsity structure of the numerator relationship matrix, and formulate the
SOCP constraint using Henderson's algorithm. Numerical results show that the
proposed SOCP approach reduced computation time in a case study from 39,200
seconds under the SDP approach to less than 2 seconds.
