Investigating the existence of function roots using particle swarm optimization

The existence of roots of functions is a topic of major significance in nonlinear analysis, and it is directly related to the problem of detection of extrema of a function. The topological degree of a function is a mathematical tool of great importance for investigating the existence and the number of roots of a function with certainty. For the computation of the topological degree according to Stenger's theorem, a sufficient refinement of the boundary of the polyhedron under consideration is needed. The sufficient refinement can be computed using the optimal complexity algorithm of Boult and Sikorski. However, the application of this algorithm requires the computation of the infinity norm on the boundary of the polyhedron under consideration as well as an estimation of the Lipschitz constant of the function. We introduced a new technique for the computation of the infinity norm on the polyhedron's boundary as well as for the estimation of the Lipschitz constant. The proposed approach is illustrated on several test problems and the results are reported and discussed.