Pattern discrete and mixed Hit‐and‐Run for global optimization

We develop new Markov chain Monte Carlo samplers for neighborhood generation in global optimization algorithms based on Hit‐and‐Run. The success of Hit‐and‐Run as a sampler on continuous domains motivated Discrete Hit‐and‐Run with random biwalk for discrete domains. However, the potential for efficiencies in the implementation, which requires a randomization at each move to create the biwalk, lead us to a different approach that uses fixed patterns in generating the biwalks. We define Sphere and Box Biwalks that are pattern‐based and easily implemented for discrete and mixed continuous/discrete domains. The pattern‐based Hit‐and‐Run Markov chains preserve the convergence properties of Hit‐and‐Run to a target distribution. They also converge to continuous Hit‐and‐Run as the mesh of the discretized variables becomes finer, approaching a continuum. Moreover, we provide bounds on the finite time performance for the discrete cases of Sphere and Box Biwalks. We embed our samplers in an Improving Hit‐and‐Run global optimization algorithm and test their performance on a number of global optimization test problems.