Conic relaxation approaches for equal deployment problems

Abstract

An important problem in the breeding of livestock, crops, and forest trees is the optimum selection of genotypes that maximize genetic gain. The key constraint in the optimal selection is a convex quadratic constraint that ensures genetic diversity, so that optimal selection can be cast as a second-order cone programming (SOCP) problem. Yamashita et al. (2015) exploit the structural sparsity of the quadratic constraints and reduce the computation time drastically while attaining the same optimal solution.This paper is concerned with a special case of equal deployment (ED), in which we solve the optimal selection problem with the constraint that contribution of genotypes must either be a fixed size or zero. This involves a form of combinatorial optimization, and the ED problem can be described as a mixed-integer SOCP problem.In this paper, we discuss conic relaxation approaches for the ED problem based on LP (linear programming), SOCP, and SDP (semidefinite programming). We then propose a steepest-ascent method that combines the solution obtained from the conic relaxation with a concept from discrete convex optimization, in order to acquire an approximate solution for the ED problem in a practical time. From numerical tests, we observed that among the LP, SOCP, and SDP relaxation problems, SOCP gave a suitable solution from the viewpoint of the optimality and computation time. The steepest-ascent method starting from the SOCP solution provides high-quality solutions much faster than other existing methods that have been widely used for optimal selection.