## The Annals of Statistics

- Ann. Statist.
- Volume 6, Number 6 (1978), 1273-1285.

### The Convergence of General Step-Length Algorithms for Regular Optimum Design Criteria

#### Abstract

For a regular optimality criterion function $\Phi$, a sequence of design measures $\{\xi_n\}$ is generated using the iteration $\xi_{n+1} = (1 - \alpha_n)\xi_n + \alpha_n\xi_n$, where $\xi_n$ is chosen to minimize $\nabla \Phi(M(\xi_n), M(\xi))$ over all $\xi$ and $\{\alpha_n\}$ is a prescribed sequence of numbers from (0, 1). This is called a general step-length algorithm for $\Phi$. Typical conditions on $\{\alpha_n\}$ are $\alpha_n \rightarrow 0$ and $\Sigma_n\alpha_n = \infty$. In this paper, a dichotomous behavior of $\{\xi_n\}$ is proved under the above conditions on $\{\alpha_n\}$ for $\Phi$ satisfying some mild regularity conditions. Sufficient conditions for convergence to optimal designs are also established. This can be applied to show that the $\{\xi_n\}$ as constructed above do converge to an optimal design for most of the trace-related and determinant-related design criteria.

#### Article information

**Source**

Ann. Statist., Volume 6, Number 6 (1978), 1273-1285.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176344373

**Digital Object Identifier**

doi:10.1214/aos/1176344373

**Mathematical Reviews number (MathSciNet)**

MR523762

**Zentralblatt MATH identifier**

0396.62059

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62K05: Optimal designs

Secondary: 62L05: Sequential design

**Keywords**

General step-length algorithms optimal design algorithms convex programming $D$-optimality $A$-optimality $\phi_p$-optimality $D_s$-optimality asymptotic convergence

#### Citation

Wu, Chien-Fu; Wynn, Henry P. The Convergence of General Step-Length Algorithms for Regular Optimum Design Criteria. Ann. Statist. 6 (1978), no. 6, 1273--1285. doi:10.1214/aos/1176344373. https://projecteuclid.org/euclid.aos/1176344373