New proposals for modelling and solving the problem of covering solids using spheres of different radii

Abstract

Given a solid T , represented by a compact set in R 3 , the aim of this work is to find a covering of T by a finite set of spheres of different radii. Some level of intersection between the spheres is necessary to cover the solid. Moreover, the volume occupied by the spheres on the outside of T is limited. This problem has an application in the planning of a radio-surgery treatment known by Gamma Knife and can be formulated as a non-convex optimization problem with quadratic constraints and linear objective function. In this work, two new linear mathematical formulations with binary variables and a hybrid method are proposed. The hybrid method combines heuristic, data mining and an exact method. Computational results show that the proposed methods outperform the ones presented in the literature.Mathematics Subject Classification. 90C10.

Designing screen layout in multimedia applications through integer programming and metaheuristic

“…Soft computing techniques are being used to deal with many layout optimization problems [15; 40]. The constructive heuristic applied here uses the black-box LocalSolver framework 2 , which has been used successfully in covering problems [15]. The LocalSolver framework has a hybrid approach of neighborhood search, which allows it to combine different optimization techniques in a dynamic way during the resolution process.…”