When LINGO finds a solution to a linear optimization model, it is the definitive best solution¾ we say it is the global optimum. A globally optimal solution is a feasible solution with an objective value that is as good or better than all other feasible solutions to the model. The ability to obtain a globally optimal solution is attributable to certain properties of linear models.

This is not the case for nonlinear optimization. Nonlinear optimization models may have several solutions that are local optimums. All nonlinear solvers converge to a locally optimal point. That is, a solution for which no better feasible solutions can be found in the immediate neighborhood of the given solution. Although better solutions can't be found in the immediate neighborhood of the local optimum, additional local optimums may exist some distance away from the current solution. These additional locally optimal points may have objective values substantially better than the solver's current local optimum. Thus, when a nonlinear model is solved, we say the solution is merely a local optimum, and the user must be aware other local optimums may, or may not, exist with better objective values.

Consider the following small nonlinear model involving the highly nonlinear cosine function:

MIN = X * @COS( 3.1416 * X);

X < 6;

(Note, graph not shown. Go to Lingo and click Help Topics. Select the “Index” Window and open Local Optima topic)

The following graph shows a plot of the objective function for values of X between 0 and 6. You can see, if you're searching for a minimum, there are local optimums at X values of 0, 1.09, 3.03, and 5.02 in the "valleys". The global optimum for this problem is at X = 5.02, because it is the lowest, feasible valley.

Graph of X * @COS( 3.1416 * X)

Imagine the graph as a series of hills. You're searching in the dark for the minimum or lowest elevation. If you are at X = 2.5, every step towards 2 takes you uphill and every step towards 3 takes you downhill. Therefore, you move towards 3 in your search for the lowest point and you'll continue to move in direction as long as it leads to lower ground.
When you reach X=3.03, you'll notice a small flat area (slope is equal to zero). Continuing begins to lead uphill and retreating leads up the hill you just came down. You're in a valley, the lowest point in the immediate neighborhood¾a local optimum. However, is it the lowest possible point? In the dark, you are unable to answer this question with certainty.

LINGO, and all solvers, face similar difficulties in solving such nonlinear models. LINGO lets you enter initial values for variables using an INIT section—the point from which LINGO begins its search—so, if you solve a model with different initial values for X, LINGO may return a different local optimum. In this example, one might imagine that starting at a value of X=6, for example, would lead to the global optimum at X=5.02. Unfortunately, this is not guaranteed, because LINGO approximates the true underlying nonlinear functions using linear and/or quadratic functions. In the early stages of solving the model, these approximations can be somewhat rough in nature and, therefore, can send the solver off to distant points, effectively passing over nearby local optimums to the true, underlying model.

In many situations, you will know "good" initial values for the variables. In this case, you should input these values in an INIT section and, additionally, you may want to use the @BND function to bound the variables within a reasonable neighborhood of the starting point. In the case where you have no idea what the optimal solution to your model is, you may find observing the results of solving the model several times with different initial values can help you find the best solution.