Limitations on Nonlinear Problems
Nonlinear problems are intrinsically more difficult to solve than linear problems, and there are fewer guarantees about what the Solver (or any optimization method) can do. Whenever the "GRG Solver" choice appears in the Solver dropdown list in the Solver Parameters dialog, the GRG (Generalized Reduced Gradient) algorithm is used to solve the problem -- even if it is actually a linear model that could be solved by the (faster and more reliable) Simplex method. The GRG method will usually find the optimal solution to a linear problem -- but occasionally you will receive a Solver Completion Message indicating some uncertainty about the status of the solution -- especially if the model is poorly scaled, as discussed above. So you should always ensure that you have selected the right Solver "engine" for your problem.
When dealing with a nonlinear problem, it is a good idea to run the Solver starting from several different sets of initial values for the decision variables. Since the Solver follows a path from the starting values (guided by the direction and curvature of the objective function and constraints) to the final solution values, it will normally stop at a peak or valley closest to the starting values you supply. By starting from more than one point -- ideally chosen based on your own knowledge of the problem -- you can increase the chances that you have found the best possible "optimal solution."
If your model has certain mathematical properties, such as convexity, it is possible to make stronger guarantees about the Solver's ability to find the true optimal solution. For more information on this topic consult the books recommended in the Introduction, particularly the titles by Wayne Winston.
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