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Sena Safarina(safarina.s.aam.titech.ac.jp) Abstract: An important problem in the breeding of livestock, crops, and forest trees is the optimum of selection of genotypes that maximizes genetic gain. The key constraint in the optimal selection is a convex quadratic constraint that ensures genetic diversity, therefore, the optimal selection can be cast as a secondorder cone programming (SOCP) problem. Yamashita et al. (2015) exploits the structural sparsity of the quadratic constraints and reduces the computation time drastically while attaining the same optimal solution. This paper is concerned with the 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 nature of combinatorial optimization, and the ED problem can be described as a mixedinteger 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 analyze theoretical bounds derived from the SDP relaxation approaches using the work of Tseng (2003) and show that the theoretical bounds are not quite sharp for tree breeding problems. We propose a steepestascent method that combines the solution obtained from the conic relaxation problems 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 viewpoints of the optimal values and the computation time. The steepestascent method starting from the SOCP solution provides highquality solutions much faster than an existing method that has been widely used for the optimal selection problems and a branchandbound method. Keywords: Semidefinite Programming; Secondorder cone programming; Mixedinteger conic programming; Conic Relaxation; Tree Breeding; Equal development problem Category 1: Linear, Cone and Semidefinite Programming Category 2: Integer Programming ((Mixed) Integer Nonlinear Programming ) Citation: Report Number: B485 Department of Mathematical and Computing Sciences Tokyo Institute of Technology March 2017 Download: [PDF] Entry Submitted: 03/07/2017 Modify/Update this entry  
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