In this chapter, we consider constraints that are neither bounds nor masks. If there are many constraints, then the problem is usually considered mathematical programming, see Chapter 14, and it is common that the constraints are then often linear. There may also be other features, such as variables that are all integers, or there is a mix of integer and real parameters. Such problems have generally been less common in statistics, but there appears to be growing interest.
In this chapter, however, we consider problems with just a few constraints, so that the main focus of attention is still the objective function.
Equality constraints are, in my experience, usually more troublesome than inequality constraints. This might seem paradoxical. After all, each equality constraint should reduce the dimension of the problem by one parameter. The difficulty is in choosing WHICH parameter to eliminate.
Nash (1979, p. 184) presents a problem (with seemingly erroneous results reported) that is apparently a linear regression except that four of the coefficients are related. We have six variables and want a model
However, there is a constraint so that
In this example, it is very obvious that we can solve for any one ...