Bounds, sometimes called box constraints, are among the most common constraints that users wish to impose on nonlinear optimization or modeling parameters. Fortunately, they are also quite easy to include, although as always care is needed in their implementation and use.
In this treatment, I will generally use the term “bounds.” In fact, we will not always impose a lower and upper bounds on every parameter, so we will not always have a “box,” and some methods do not lend themselves to imposing a true -dimensional box in which we will seek a solution.
The general conditions we wish to satisfy are, therefore,
Suppose we want one of our optimization parameters—call it —to be nonnegative in our modeling or objective function . For the moment, we will ignore other inputs to , and we will leave the generalization to bounds other than zero until later. Clearly, we could rewrite our function to use a different ...