Global Optimization Toolbox for Maple

106 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006 1066-033X/06/$20.00©2006IEEE Global optimization seeks to find the best solution to a constrained nonlinear optimization problem by performing a complete search over a set of feasible solutions. In contrast with local optimization, a complete search exhaustively checks the entire feasible region under a Lipschitz-continuity assumption with a known bound on the Lipschitz constant. Some online information on global optimization is available at [1]. As surveyed in [2], many mathematical and engineering problems require a complete search. An example is the 300year-old Kepler problem of finding the densest packing of equal spheres in three-dimensional Euclidean space, for which a computer-assisted proof is discussed in [3]. The proof involves reducing the problem to several thousand linear programs and using interval calculations to ensure rigorous handling of rounding errors for establishing the correctness of inequalities. Many other difficult problems, such as the traveling salesman and protein folding problems, are global optimization problems.