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.

We consider Stackelberg patrolling security games in which a security guard and an intruder walk around a facility. In these games, at each timepoint, the guard earns a reward (intruder incurs a cost) depending on their locations at that time. The objective of the guard (resp., the intruder) is to patrol (intrude) the facility so that the total sum of rewards is maximized (minimized). We study three cases: In Case 1, the guard chooses a scheduled route first and then the intruder chooses a scheduled route after perfectly observing the guard's choice. In Case 2, the guard randomizes her scheduled routes and then intruder observes its probability distribution and also randomize his scheduled routes. In Case 3, the guard randomizes her scheduled routes as well, but the intruder sequentially observes the location of the guard and reroutes to reach one of his targets. We show that the intruder's best response problem in Cases 1 and 2 and Case 3 can be formulated as a shortest path problem and a Markov decision process, respectively. Moreover, the equilibrium problem in each case reduces to a polynomial-sized mixed integer linear programming, linear programming, and bilinear programming problem, respectively.