Computing cc-optimal experimental designs using the simplex method of linear programming

An experimental design is said to be cc-optimal if it minimizes the variance of the best linear unbiased estimator of cTβ, where cc is a given vector of coefficients, and ββ is an unknown vector parameter of the model in consideration. For a linear regression model with uncorrelated observations and a finite experimental domain, the problem of approximate cc-optimality is equivalent to a specific linear programming problem. The most important consequence of the linear programming characterization is that it is possible to base the calculation of cc-optimal designs on well-understood computational methods. In particular, the simplex algorithm of linear programming applied to the problem of cc-optimality reduces to an exchange algorithm with different pivot rules corresponding to specific techniques of selecting design points for exchange. The algorithm can also be applied to “difficult” problems with singular cc-optimal designs and relatively high dimension of ββ. Moreover, the algorithm facilitates identification of the set of all the points that can support some cc-optimal design. As an example, optimal designs for estimating the individual parameters of the trigonometric regression on a partial circle are computed.