Locally Versus Globally Optimal Solutions

When the first message ("Solver found a solution") appears, it means that the GRG Solver has found a locally optimal solution -- there is no other set of values for the decision variables close to the current values which yields a better value for the objective. Figuratively, this means that the Solver has found a "peak" (if maximizing) or "valley" (if minimizing) -- but there may be other taller peaks or deeper valleys far away from the current solution. Mathematically, this message means that the Karush - Kuhn - Tucker (KKT) conditions for local optimality have been satisfied (to within a certain tolerance, related to the Precision setting in the Solver Options dialog).

Although there is ongoing research in globally optimal nonlinear programming methods which we are monitoring, the best that current nonlinear methods can guarantee is to find a locally optimal solution. We recommend that you run the GRG Solver starting from several different sets of initial values for the decision variables -- ideally chosen based on your own knowledge of the problem. In this way you can increase the chances that you have found the best possible "optimal solution."