Polyhedral-based methods for mixed-integer SOCP in tree breeding

Optimal contribution selection (OCS) is a mathematical optimization problem that aims to maximize the total benefit from selecting a group of individuals under a constraint on genetic diversity. We are specifically focused on OCS as applied to forest tree breeding, where selected individuals will contribute equally to the gene pool. Since the diversity constraint in OCS can be described with a second-order cone, equal deployment in OCS can be mathematically modeled as mixed-integer second-order cone programming (MI-SOCP). However, if we apply a general solver for MI-SOCP, non-linearity embedded in OCS requires a heavy computation cost. To address this problem, we propose an implementation of lifted polyhedral programming (LPP) relaxation and a cone-decomposition method (CDM) by generating effective linear approximations for OCS. Furthermore, to enhance the performance of CDM, we utilize the sparsity structure that can be discovered in OCS. Through numerical experiments, we verified CDM with the sparse structure successfully solves OCS problems much faster than generic approaches for MI-SOCP.