The purpose of this paper is to introduce a new way of choosing directions for the Mesh Adaptive Direct Search (MADS) class of algorithms. The advantages of this new ORTHOMADS instantiation of MADS are that the polling directions are chosen deterministically, ensuring that the results of a given run are repeatable, and that they are orthogonal to each other, therefore the convex cones of missed directions at each iteration are minimal in size. The convergence results for ORTHOMADS follow directly from those already published for MADS, and they hold deterministically, rather than with probability one, as for LTMADS, the first MADS instance. The initial numerical results are quite good for both smooth and nonsmooth, and constrained and unconstrained problems considered here.

Abstract: We consider derivative-free optimization, and in particular black box optimization, where the functions to minimize and the functions representing the constraints are given by black boxes without derivatives. Two fundamental families of methods are available: Model-based methods and directional direct search algorithms. This work exploits the flexibility of the second type of method in order to integrate to a limited extent the models used in the first family. Intensive numerical tests on two sets of forty-eight and one hundred and four test problems illustrate the efficiency of this hybridization and show that the use of models improves significantly the mesh adaptive direct search algorithm.

Abstract: The Mesh Adaptive Direct Search (MADS) class of algorithms is designed for nonsmooth optimization, where the objective function and constraints are typically computed by launching a time-consuming computer simulation. Each iteration of a MADS algorithm attempts to improve the current best-known solution by launching the simulation at a finite number of trial points. Common implementations of MADS generate 2n trial points at each iteration, where n is the number of variables in the optimization problem. The objective of the present work is to reduce that number. We present an algorithmic framework that reduces the number of simulations to exactly n + 1, without impacting the theoretical guarantees from the convergence analysis. Numerical experiments are conducted for several different contexts; the results suggest that these strategies allow the new algorithms to reach a better solution with fewer function evaluations.

Abstract: Accurate measurements of snow water equivalent (SWE) is an important factor in managing water resources for hydroelectric power generation. SWE over a catchment area may be estimated via kriging on measures obtained by snow monitoring devices positioned at strategic locations. The question studied in this paper is to find the device locations that minimize the kriging interpolation error of the SWE. This is done first by formulating a simulator blackbox that takes a set of locations as inputs and returns the interpolation error, and then to minimize this error using the mesh adaptive direct search (MADS) algorithm designed for blackbox optimization. The fact that the optimization variables represent planar coordinates is used to devise algorithmic strategies that dynamically groups subsets of variables. The methodology is applied to three water-resource systems in the province of Québec on the blackbox simulator and on a surrogate with various grouping strategies.