Numerical Solution of the Defence Force Optimal Positioning Problem

In this paper we study the positioning of defender forces in order to handle in an efficient way the forces of the attacker. The scope is to determine the minimum amplitude of territories which the invader will occupy. The defender’s forces should swoop rapidly to any point of the defence locus in order to protect their territories. The selection of the “optimal” position in which the defender’s forces should be placed is a difficult problem and it aims at the minimization of enemy’s penetration. The minimization methods result to non-linear equations and there are many classical numerical algorithms for solving such equations. The most known one is Newton’s method. Since the selection of a suitable initial point is not a trivial task, we will study the behaviour of these numerical procedures for various initial points and small perturbations of the data in order to present stable procedures which compute efficiently the solution of non-linear equations, leading to the optimal selection of the position, on which the forces of the defender should be placed. All the proposed methods are tested for various sets of data and useful conclusions arise. The algorithms are compared as to the computational complexity and stability through error analysis yielding useful results.