The GRG Nonlinear Solving method, like most nonlinear optimization methods, normally can find only a locally optimal solution to a nonlinear, non-convex problem.  Solver includes a multistart method that can improve your prospects of finding a globally optimal solution for such a problem.  The basic idea of the multistart method is to automatically run the GRG “local Solver” from different starting points, reaching different locally optimal solutions, then select the best of these as the proposed globally optimal solution.

The multistart method operates by generating candidate starting points for the local Solver (with randomly selected values between the bounds you specify for the variables).  These points are then grouped into “clusters” – through a method called multi-level single linkage – that are likely to lead to the same locally optimal solution.  The local Solver is then run repeatedly, once from (a representative starting point in) each cluster.  The process continues with successively smaller clusters that are increasingly likely to capture each possible locally optimal solution.  A Bayesian test is used to determine whether the process should continue or stop.

For some smooth nonlinear problems, the multistart method has a limited guarantee that it will “converge in probability” to a globally optimal solution.  This means that as the number of runs of the nonlinear Solver increases, the probability that the globally optimal solution has been found also increases towards 100%.  (To attain convergence for constrained problems, an exact penalty function is used in the process of “clustering” the starting points.)  For most nonlinear problems, this method will at least yield very good solutions.