A review of the Global Optimization Toolbox for Maple

As surveyed in [7], there are numerous mathematical and engineering problem classes for which a complete search is required. An example is the 300 year old Kepler problem of finding the densest packing of equal spheres in 3-dimensional Euclidean space, for which a computer-assisted proof is proposed by T. C. Hales [4]. The proof consists in reducing the problem to several thousands of linear programs and using interval calculations to ensure rigorous handling of rounding errors when establishing correctness of inequalities. Many other famous difficult optimization problems, such as the traveling salesman problem and the protein folding problem, are global optimization problems.