Given a starting vector for the decision variables,
the Excel Solver program uses a search procedure to move to
a local optimum. That is, a point in the feasible region such
that no improvement can be made by small perturbations from
the point. It is a known result that when maximizing a concave
objective function with linear constraints every local optimum
is a global optimum, the solution that is the best of all
feasible solutions. Since the objective function terms for
this example are concave, when the Solver program terminates
at a point with an indication of optimality, the point should
be the global optimum.
In general, we cannot be sure that the Excel
Solver or any nonlinear programming algorithm will terminate
at an optimal point unless some rather stringent conditions
are satisfied. These conditions are the subject of nonlinear
programming textbooks. For example if the objective function
terms were convex, rather than concave, the algorithm will
always terminate at an extreme point of the feasible region
(when the region is defined by linear constraints). In many
cases the extreme points will not be globally optimum solutions.
The situation is even further complicated for integer-nonlinear
models. An extensive coverage of these features would take
many pages, but the user should beware of the problem of assuring
optimality for nonlinear models.
There are limitations on the functions that
can appear in nonlinear terms. The functions should be defined,
continuous and differentiable in and near the feasible region.
This eliminates the use of IF, integer, and absolute value
functions in the model. Ratios with denominators that might
go to zero or square root terms that have negative arguments
are similarly not allowed. These restrictions must be obeyed
at every point that the Solver program might try to evaluate.
Even if the initial solution is feasible, the Solver program
might make small incursions outside the feasible region during
the search process. If the program encounters a point for
which a function cannot be evaluated it stops with an error
The problem of finding the global optimum from
perhaps a large set of local optima is been the subject of
much theoretical as well as practical effort. This field
of research and practice is called Global Optimization.
Frontline Systems, the distributor of the Excel
Solver has premium systems that implement highly advanced
methods for finding global optimum solutions.