Using semidefinite programming to optimize unequal deployment of genotypes to a clonal seed orchard

Abstract

Tree breeders must often consider the conservation of genetic diversity, while at the same time, maximizing response to selection. In the case of seed orchards, the buyer of seed wants maximum performance, while satisfying a restriction, sometimes legislated, on the diversity deployed to the forest. Optimal selection will not completely avoid kinship but rather maximize gain while imposing a constraint on average relatedness. Here, we present the application of semidefinite programming (SDP) as a flexible approach to optimize the deployment of genotypes to a clonal seed orchard. We formulate the selection problem as an SDP, where average breeding value is to be maximized, while imposing constraints on relatedness, as well as maximum and minimum contributions from each candidate. An open-source solver, SDPA, was embedded into a tool designed to make the optimization of seed orchards by SDP simple and flexible. Case studies optimizing seed orchards for Scots pine and loblolly pine illustrate how this flexibility can be used to impose additional constraints on the scion material available from some candidate genotypes and optimize selection even when related candidates have varying degrees of coancestry among them. Additional situations where SDP can be employed are discussed. © 2013 The Author(s).