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The Nelder-Mead or simplex search algorithm is one of the best known algorithms for unconstrained optimization of non-smooth functions. Even though the basic algorithm is quite simple, it is implemented in many different ways. Apart from some minor computational details, the main difference between various implementations lies in the selection of convergence (or termination) tests, which are used to break the iteration process.A fairly simple efficiency analysis of each iteration step reveals a potential computational bottleneck in the domain convergence test. To be efficient, such a test has to be sublinear in the number of vertices of the working simplex. We have tested some of the most common implementations of the Nelder-Mead algorithm, and none of them is efficient in this sense.Therefore, we propose a simple and efficient domain convergence test and discuss some of its properties. This test is based on tracking the volume of the working simplex throughout the iterations. Similar termination tests can also be applied in some other simplex-based direct search methods.

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A one-sided Jacobi hyperbolic singular value decomposition (HSVD) algorithm, using a massively parallel graphics processing unit (GPU), is developed. The algorithm also serves as the final stage of solving a symmetric indefinite eigenvalue problem. Numerical testing demonstrates the gains in speed and accuracy over sequential and MPI-parallelized variants of similar Jacobi-type HSVD algorithms. Finally, possibilities of hybrid CPU--GPU parallelism are discussed.Comment: Accepted for publication in BIT Numerical Mathematic

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