Abstract. Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis. Nonetheless, users remained loyal to these methods, most of which were easy to program, some of which were reliable. In the past fifteen years, these methods have seen a revival due, in part, to the appearance of mathematical analysis, as well as to interest in parallel and distributed computing. This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited. Our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results. The underlying principles allow generalization to handle bound constraints and linear constraints. We also discuss extensions to problems with nonlinear constraints.

We introduce an abstract de nition of pattern search methods for solving nonlinear unconstrained optimization problems. Our de nition uni es an important collection of optimization methods that neither computenor explicitlyapproximatederivatives. We exploitour characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of su cient decrease. Our analysis is possible because the iterates of a pattern search method lie on a scaled, translated integer lattice. This allows us to relax the classical requirements on the acceptance of the step, at the expense of stronger conditions on the form of the step, and still guarantee global convergence.

The goal of the research reported here is to develop rigorous optimization algorithms to apply to some engineering design problems for which direct application of traditional optimization approaches is not practical. This paper presents and analyzes a framework for generating a sequence of approximations

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