Algorithms

Abstract

Theorem 3.7 was used in Example 3.8 to verify that an intuitively sensible design measure was in fact D-optimal. However to exploit its full potential we require more than this; we need algorithms that enable us to construct Φ-optimal design measures. Now we know that there always exists an optimal measure with finite support. Suppose that we know a priori a finite collection of points x₍₁₎,…,x_(T), say, among which the support points of a Φ-optimal measure lie. Then the only problem is to find the appropriate probabilities at these points. This is a standard extremum problem; we have a concave function $f\left( {{\eta _1},\,\, \ldots {\eta _T}} \right) = \phi \left( {\sum {{\eta _i}{x_{\left( i \right)}}x_{\left( i \right)}^T} } \right)$ that we wish to maximize subject to ${\eta _i} \geqslant 0,\sum {{\eta _i} = 1}$; and it can be tackled by standard numerical methods; see Wu (1978); also Silvey, Titterington & Torsney (1978) for a purpose-built algorithm.