Abstract. This paper introduces the Mesh Adaptive Direct Search (MADS) class of algorithms for nonlinear optimization. MADS extends the Generalized Pattern Search (GPS) class by allowing local exploration, called polling, in a dense set of directions in the space of optimization variables. This means that under certain hypotheses, including a weak constraint qualification due to Rockafellar, MADS can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained. The main GPS convergence result is to identify limit points where the Clarke generalized derivatives are nonnegative in a finite set of directions, called refining directions. Although in the unconstrained case, nonnegative combinations of these directions spans the whole space, the fact that there can only be finitely many GPS refining directions limits rigorous justification of the barrier approach to finitely many constraints for GPS. The MADS class of algorithms extend this result; the set of refining directions may even be dense in R n , although we give an example where it is not.We present an implementable instance of MADS, and we illustrate and compare it with GPS on some test problems. We also illustrate the limitation of our results with examples.

This paper contains a new convergence analysis for the Lewis and Torczon generalized pattern search (GPS) class of methods for unconstrained and linearly constrained optimization. This analysis is motivated by a desire to understand the successful behavior of the algorithm under hypotheses that are satisfied by many practical problems. Specifically, even if the objective function is discontinuous or extended valued, the methods find a limit point with some minimizing properties. Simple examples show that the strength of the optimality conditions at a limit point does not depend only on the algorithm, but also on the directions it uses, and on the smoothness of the objective at the limit point in question. The contribution of this paper is to provide a simple convergence analysis that supplies detail about the relation of optimality conditions to objective smoothness properties and to the defining directions for the algorithm, and it gives previous results as corollaries.

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This paper formulates and analyzes a pattern search method for general constrained optimization based on filter methods for step acceptance. Roughly, a filter method accepts a step that improves either the objective function value or the value of some function that measures the constraint violation. The new algorithm does not compute or approximate any derivatives, penalty constants, or Lagrange multipliers. A key feature of the new algorithm is that it preserves the division into search and local poll steps, which allows the explicit use of inexpensive surrogates or random search heuristics in the search step. It is shown here that the algorithm identifies limit points at which optimality conditions depend on local smoothness of the functions and, to a greater extent, on the choice of a certain set of directions. Stronger optimality conditions are guaranteed for smoother functions and, in the constrained case, for a fortunate choice of the directions on which the algorithm depends. These directional conditions generalize those given previously for linear constraints, but they do not require a feasible starting point. In the absence of general constraints, the proposed algorithm and its convergence analysis generalize previous work on unconstrained, bound constrained, and linearly constrained generalized pattern search. The algorithm is illustrated on some test examples and on an industrial wing planform engineering design application.

A new class of algorithms for solving nonlinearly constrained mixed variable optimization problems is presented. This class combines and extends the Audet-Dennis generalized pattern search (GPS) algorithms for bound constrained mixed variable optimization, and their GPS-filter algorithms for general nonlinear constraints. In generalizing existing algorithms, new theoretical convergence results are presented that reduce seamlessly to existing results for more specific classes of problems. While no local continuity or smoothness assumptions are required to apply the algorithm, a hierarchy of theoretical convergence results based on the Clarke calculus is given, in which local smoothness dictates what can be proved about certain limit points generated by the algorithm. We believe this is the first algorithm with provable convergence results to directly target this class of problems.