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

An efficient second-order cone programming approach for optimal selection in tree breeding