Monotonic convergence of a general algorithm for computing optimal designs

Monotonic convergence is established for a general class of multiplicative algorithms introduced by Silvey, Titterington and Torsney [Comm. Statist. Theory Methods 14 (1978) 1379–1389] for computing optimal designs. A conjecture of Titterington [Appl. Stat. 27 (1978) 227–234] is confirmed as a consequence. Optimal designs for logistic regression are used as an illustration.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of multiplicative algorithms for optimal experimental design. Since the seminal work of Silvey, Titterington and Torsney (1978), a class of update rules—often called “multiplicative algorithms”—has been widely used to compute designs that optimize a variety of criteria (D‑optimality, A‑optimality, etc.). Although these algorithms have been observed to converge in practice, a rigorous proof of monotonic convergence—i.e., that the objective function never worsens from one iteration to the next—has been missing. Moreover, Titterington (1978) conjectured that every initial design would converge monotonically to a globally optimal design, but this conjecture remained unproven.

The authors first formalize the problem. Let ξ^{(t)} = { (x_i, w_i^{(t)}) }_{i=1}^n denote the design at iteration t, where w_i^{(t)} are non‑negative weights summing to one and A_i is the information matrix contributed by point x_i. The overall information matrix is M(ξ^{(t)}) = Σ_i w_i^{(t)} A_i. The design criterion Φ(M) is assumed to be convex and homogeneous of degree –p (p>0), a property satisfied by most classical optimality criteria.

A generalized multiplicative update is introduced: \