Constrained Maximisation and the Design of Experiments

The February 1968 issue of Technometrics contained two papers in which the problem of maximising (or minimising) a function of several variables arose, where the values of the variables were subject to constraints. It is the purpose of this note to call attention to the use of transformations whereby certain forms of such constraints can be eliminated. The method was suggested by M. J. Box [1]. His paper does not seem to be well known to statisticians. It is hoped that what follows will serve to publicise this useful technique. With one exception, all the transformations in the present paper are given by Box. The paper by Hill, Hunter and Wichern [5] is an example of this situation in the context of the design of experiments. The constraints on the process variables time and temperature define a region of operability. The design problem is to find the maximum in this region of a function of the process variables called the design criterion. For one or two variables the maximum may easily be found by searching over a grid. For more variables, however, this process rapidly becomes inefficient. In an unconstrained problem recourse would be made to one of the well established hill climbing programmes such as that of Powell [8]. Use of such a procedure when constraints are present may well result in the location of a maximum outside the region of operability. In such a situation the use of transformations may allow us to employ the power of the hill climbing technique whilst ensuring that the solution obtained does not violate any of the constraints. As the simplest example, suppose we have a variable xi which is required to be non negative. Then if we write xi = y2 , an unconstrained search over all values of y, is equivalent to a constrained search for xi . The next simplest problem arises in the design of experiments in regression situations. See, for example, Clark [2]. Here each variable is subject to the constraint -1 < xi < 1, and the design criterion is to maximise the value of the