Informatica

We investigate the problem of detecting a point set’s deviation from uniformity in the unit hypercube. High uniformity is for example desirable in Monte Carlo methods for numerical integration, but also for obtaining a good worst-case bound in global optimization. In high dimensions, many points are required to get reliable results, so the point sets are preferably generated by fast methods such as quasirandom sequences. Unfortunately, assessing their uniformity often requires quadratic time. So, we present several numerical summary characteristics of point sets that can be computed in linear time. They do not measure uniformity directly, but by comparing them to reference values for the uniform distribution, deviations from uniformity can be quickly detected. The necessary reference values are also derived here, if possible exactly, else approximately.

This work is our first attempt in establishing the connections between evolutionary computation algorithms and stochastic approximation procedures. By treating evolutionary algorithms as recursive stochastic procedures, we study both constant gain and decreasing step size algorithms. We formulate the problem in a rather general form, and supply the sufficient conditions for convergence (both with probability one, and in the weak sense). Among other things, our approach reveals the natural connection of the discrete iterations and the continuous dynamics (ordinary differential equations, and/or stochastic differential equations). We hope that this attempt will open up a new horizon for further research and lead to in depth understanding of the underlying algorithms.