Statistica Sinica 7(1997), 739-753 COMPUTING OPTIMAL DESIGNS BY BUNDLE TRUST METHODS Adalbert Wilhelm Universit¨atAugsburg Abstract: Essentially two classes of iterative procedures have been proposed in the literature to solve optimal design problems in linear regression: exchange algorithms devoted to the construction of optimal exact designs in a finite design space and methods from convex programming yielding optimal moment matrices only. By simultaneously taking weights and support points as variables the design problem represents a nonconcave, not necessarily differentiable, but Lipschitz continuous maximization problem. We, therefore, adapt bundle trust methods from nondiffer- entiable optimization to the design problem and show their numerical behaviour. Explicit efficiency bounds for the numerical solutions can be given in the case of a regression range with finitely many elements. Key words and phrases: Approximate designs, bundle trust methods, nondifferen- tiability, pth matrix means, support points and weights. 1. Introduction Algorithms for computing optimal experimental designs traditionally fall in one of the following two classes: the class of exchange algorithms for constructing optimal designs for a finite design space and the class of convex programming methods that only yield optimum moment matrices leaving the problem how to reconstruct a design from a given moment matrix unsolved. The first versions of exchange algorithms were addressed to D-optimality and go back to Wynn (1970) and Fedorov (1972). Reviews of these and other exchange schemes are given by Cook and Nachtsheim (1980) and, more recently, by Nguyen and Miller (1992). A new approach to the construction of optimal exact designs using pattern search was proposed by Hardin and Sloane (1993). Gaffke and Mathar (1992) gave an excellent review of convex programming algorithms used in the design context. In the present article we introduce a new algorithm for computing optimal or nearly optimal approximate experimental designs. This algorithm is based on the bundle trust method developed by Schramm and Zowe (1988). Bundle methods have become popular in nondifferentiable convex optimization in the 1980’s (see Hiriart-Urruty and Lemar´ echal (1993) for a thorough introduction).