Is Differential Evolution Rotationally Invariant?

In this paper, we study a problem of the control parameter settings in Differential Evolution algorithm and test a novel variant of the algorithm called CoBiDE. Although Differential Evolution with basic setting (i.e., CR=0.5; F =0.5) works quite well, it starts to fail on rotated functions. In general, we want to improve the convergence of algorithm primarily on rotated functions. It is done by adapting crossover parameter CR whereas parameter F is fixed to 0.5. There is a recommendation to set CR = 1 for rotated functions. It means that trial vectors are essentially composed from mutant. However, it is not easy task to set the parameters appropriately for solving optimization problem but it is crucial for obtaining good results. Moreover, the quality of points produced in evolution is highly affected by the coordinate system. In CoBiDE, the authors proposed a new coordinate system based on the current distribution of points in the population. We test these two approaches by running both algorithms on six pairs of rotated and non-rotated functions from CEC 2013 benchmark set in two levels of dimension space. This experimental study aims to reveal if such algorithm’s setting is invariant under a rotation.

LATTICE EQUILIBRIUM THEORY AND SIZE EFFECTS FOR BOSONS IN A BOUNDED HARMONIC POTENTIAL

Effects of finite spatial size of boson assemblies in traps are studied in a self-consistent lattice theory by modeling the trap as a bounded harmonic potential of size R0. The thermodynamic quantities exhibit scaling and crossover from ideal gas behaviour at small (R0/a0) to that appropriate to an unbounded harmonic potential at large (R0/a0) with a crossover parameter [Formula: see text], a0 being the harmonic oscillator length, and τ denoting the dimensionless thermal energy. The numerical results obtained earlier by computing the energy levels of the bounded harmonic oscillator fit the general structure predicted by the theory very well. For a1>10, the spatial size effects are negligible but for a1<10 they become appreciable and experimentally measurable in suitably designed traps. At low temperatures the self consistent cell size is found to be about 2.5a0 implying that the condensate is essentially a single coherent state contained in the central cell.