The perimeter of uniform and geometric words: a probabilistic analysis

Let a word be a sequence of $n$ i.i.d. integer random variables. The perimeter $P$ of the word is the number of edges of the word, seen as a polyomino. In this paper, we present a probabilistic approach to the computation of the moments of $P$. This is applied to uniform and geometric random variables. We also show that, asymptotically, the distribution of $P$ is Gaussian and, seen as a stochastic process, the perimeter converges in distribution to a Brownian motionComment: 13 pages, 7 figure

Cumulative Step-size Adaptation on Linear Functions

The CSA-ES is an Evolution Strategy with Cumulative Step size Adaptation, where the step size is adapted measuring the length of a so-called cumulative path. The cumulative path is a combination of the previous steps realized by the algorithm, where the importance of each step decreases with time. This article studies the CSA-ES on composites of strictly increasing functions with affine linear functions through the investigation of its underlying Markov chains. Rigorous results on the change and the variation of the step size are derived with and without cumulation. The step-size diverges geometrically fast in most cases. Furthermore, the influence of the cumulation parameter is studied.Comment: arXiv admin note: substantial text overlap with arXiv:1206.120

Markov Chain Analysis of Cumulative Step-size Adaptation on a Linear Constrained Problem

This paper analyzes a (1, $\lambda$)-Evolution Strategy, a randomized comparison-based adaptive search algorithm, optimizing a linear function with a linear constraint. The algorithm uses resampling to handle the constraint. Two cases are investigated: first the case where the step-size is constant, and second the case where the step-size is adapted using cumulative step-size adaptation. We exhibit for each case a Markov chain describing the behaviour of the algorithm. Stability of the chain implies, by applying a law of large numbers, either convergence or divergence of the algorithm. Divergence is the desired behaviour. In the constant step-size case, we show stability of the Markov chain and prove the divergence of the algorithm. In the cumulative step-size adaptation case, we prove stability of the Markov chain in the simplified case where the cumulation parameter equals 1, and discuss steps to obtain similar results for the full (default) algorithm where the cumulation parameter is smaller than 1. The stability of the Markov chain allows us to deduce geometric divergence or convergence , depending on the dimension, constraint angle, population size and damping parameter, at a rate that we estimate. Our results complement previous studies where stability was assumed.Comment: Evolutionary Computation, Massachusetts Institute of Technology Press (MIT Press): STM Titles, 201

MoMA-LigPath: A web server to simulate protein-ligand unbinding

Protein-ligand interactions taking place far away from the active site, during ligand binding or release, may determine molecular specificity and activity. However, obtaining information about these interactions with experimental or computational methods remains difficult. The computational tool presented in this paper, MoMA-LigPath, is based on a mechanistic representation of the molecular system, considering partial flexibility, and on the application of a robotics-inspired algorithm to explore the conformational space. Such a purely geometric approach, together with the efficiency of the exploration algorithm, enables the simulation of ligand unbinding within very short computing time. Ligand unbinding pathways generated by MoMA-LigPath are a first approximation that can provide very useful information about protein-ligand interactions. When needed, this approximation can be subsequently refined and analyzed using state-of-the-art energy models and molecular modeling methods. MoMA-LigPath is available at http://moma.laas.fr. The web server is free and open to all users, with no login requirement