Mesh Adaptive Direct Search Algorithms for Constrained Optimization

Abstract

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.

Characteristic Shape Sequences for Measures on Images

“…Robustness of the solution to noise and experimental uncertainty is enforced by allowing the ellipses to contain a specified small percentage of pixels that lie outside the specified image level set. The optimization problem is solved numerically by applying a mesh adaptive direct search (MADS) algorithm [6] with a filter [5]. The filter allows intermediate solutions that violate constraints in order to provide a more robust global search of the parameter space.…”

“…Because of the weaknesses inherent to filter-GPS, Audet and Dennis [6] more recently introduced the class of MADS algorithms. Instead of limiting local exploration to a finite number of directions (as GPS does), MADS systematically generates an asymptotically dense set of directions in the limit.…”

“…Instead of limiting local exploration to a finite number of directions (as GPS does), MADS systematically generates an asymptotically dense set of directions in the limit. Because of this property, convergence to both first-order [6] and second-order [2] stationary points is guaranteed under reasonable assumptions without the need for a filter but with the requirement that the starting point be feasible.…”

“…Under this construction, GPS is the specific implementation of MADS for which ∆ m k = ∆ p k for all k. Figures 2 and 3 give examples of a GPS and MADS frame, respectively [6]. As shown Figure 2: Example of GPS frames…”

“…In fact, the total of all poll directions can be chosen to become asymptotically dense. A specific implementation of MADS was devised in [6] in which the sets D k are limited in size but chosen randomly at each iteration. This preserves the denseness of poll directions, but does so only with probability one.…”