Using QR Decomposition to Obtain a New Instance of Mesh Adaptive Direct Search with Uniformly Distributed Polling Directions

The purpose of this paper is to introduce a new instance of the Mesh Adaptive Direct Search (Mads) class of algorithms, which utilizes a more uniform distribution of poll directions than do other common instances, such as OrthoMads and LtMads. Our new implementation, called QrMads, bases its poll directions on an equal area partitioning of the n-dimensional unit sphere and the QR decomposition to obtain an orthogonal set of directions. While each instance produces directions which are dense in the limit, QrMads directions are more uniformly distributed in the unit sphere. This uniformity is the key to enhanced performance in higher dimensions and for constrained problems. The trade-off is that QrMads is no longer deterministic and at each iteration the set of polling directions is no longer orthogonal. Instead, at each iteration, the poll directions are only ‘nearly orthogonal,’ becoming increasingly closer to orthogonal as the mesh size decreases. Finally, we present a variety of test results on smooth, nonsmooth, unconstrained, and constrained problems and compare them to OrthoMads on the same set of problems.