An Overview of Schema Theory

The purpose of this paper is to give an introduction to the field of Schema Theory written by a mathematician and for mathematicians. In particular, we endeavor to to highlight areas of the field which might be of interest to a mathematician, to point out some related open problems, and to suggest some large-scale projects. Schema theory seeks to give a theoretical justification for the efficacy of the field of genetic algorithms, so readers who have studied genetic algorithms stand to gain the most from this paper. However, nothing beyond basic probability theory is assumed of the reader, and for this reason we write in a fairly informal style. Because the mathematics behind the theorems in schema theory is relatively elementary, we focus more on the motivation and philosophy. Many of these results have been proven elsewhere, so this paper is designed to serve a primarily expository role. We attempt to cast known results in a new light, which makes the suggested future directions natural. This involves devoting a substantial amount of time to the history of the field. We hope that this exposition will entice some mathematicians to do research in this area, that it will serve as a road map for researchers new to the field, and that it will help explain how schema theory developed. Furthermore, we hope that the results collected in this document will serve as a useful reference. Finally, as far as the author knows, the questions raised in the final section are new.Comment: 27 pages. Originally written in 2009 and hosted on my website, I've decided to put it on the arXiv as a more permanent home. The paper is primarily expository, so I don't really know where to submit it, but perhaps one day I will find an appropriate journa

SCoPE: An efficient method of Cosmological Parameter Estimation

Markov Chain Monte Carlo (MCMC) sampler is widely used for cosmological parameter estimation from CMB and other data. However, due to the intrinsic serial nature of the MCMC sampler, convergence is often very slow. Here we present a fast and independently written Monte Carlo method for cosmological parameter estimation named as Slick Cosmological Parameter Estimator (SCoPE), that employs delayed rejection to increase the acceptance rate of a chain, and pre-fetching that helps an individual chain to run on parallel CPUs. An inter-chain covariance update is also incorporated to prevent clustering of the chains allowing faster and better mixing of the chains. We use an adaptive method for covariance calculation to calculate and update the covariance automatically as the chains progress. Our analysis shows that the acceptance probability of each step in SCoPE is more than $95\%$ and the convergence of the chains are faster. Using SCoPE, we carry out some cosmological parameter estimations with different cosmological models using WMAP-9 and Planck results. One of the current research interests in cosmology is quantifying the nature of dark energy. We analyze the cosmological parameters from two illustrative commonly used parameterisations of dark energy models. We also asses primordial helium fraction in the universe can be constrained by the present CMB data from WMAP-9 and Planck. The results from our MCMC analysis on the one hand helps us to understand the workability of the SCoPE better, on the other hand it provides a completely independent estimation of cosmological parameters from WMAP-9 and Planck data.Comment: 22 pages, 10 figures, 2 table

Computationally designed libraries of fluorescent proteins evaluated by preservation and diversity of function

To determine which of seven library design algorithms best introduces new protein function without destroying it altogether, seven combinatorial libraries of green fluorescent protein variants were designed and synthesized. Each was evaluated by distributions of emission intensity and color compiled from measurements made in vivo. Additional comparisons were made with a library constructed by error-prone PCR. Among the designed libraries, fluorescent function was preserved for the greatest fraction of samples in a library designed by using a structure-based computational method developed and described here. A trend was observed toward greater diversity of color in designed libraries that better preserved fluorescence. Contrary to trends observed among libraries constructed by error-prone PCR, preservation of function was observed to increase with a library's average mutation level among the four libraries designed with structure-based computational methods

Generalized decomposition and cross entropy methods for many-objective optimization

Decomposition-based algorithms for multi-objective optimization problems have increased in popularity in the past decade. Although their convergence to the Pareto optimal front (PF) is in several instances superior to that of Pareto-based algorithms, the problem of selecting a way to distribute or guide these solutions in a high-dimensional space has not been explored. In this work, we introduce a novel concept which we call generalized decomposition. Generalized decomposition provides a framework with which the decision maker (DM) can guide the underlying evolutionary algorithm toward specific regions of interest or the entire Pareto front with the desired distribution of Pareto optimal solutions. Additionally, it is shown that generalized decomposition simplifies many-objective problems by unifying the three performance objectives of multi-objective evolutionary algorithms – convergence to the PF, evenly distributed Pareto optimal solutions and coverage of the entire front – to only one, that of convergence. A framework, established on generalized decomposition, and an estimation of distribution algorithm (EDA) based on low-order statistics, namely the cross-entropy method (CE), is created to illustrate the benefits of the proposed concept for many objective problems. This choice of EDA also enables the test of the hypothesis that low-order statistics based EDAs can have comparable performance to more elaborate EDAs

Linear Convergence of Comparison-based Step-size Adaptive Randomized Search via Stability of Markov Chains

In this paper, we consider comparison-based adaptive stochastic algorithms for solving numerical optimisation problems. We consider a specific subclass of algorithms that we call comparison-based step-size adaptive randomized search (CB-SARS), where the state variables at a given iteration are a vector of the search space and a positive parameter, the step-size, typically controlling the overall standard deviation of the underlying search distribution.We investigate the linear convergence of CB-SARS on\emph{scaling-invariant} objective functions. Scaling-invariantfunctions preserve the ordering of points with respect to their functionvalue when the points are scaled with the same positive parameter (thescaling is done w.r.t. a fixed reference point). This class offunctions includes norms composed with strictly increasing functions aswell as many non quasi-convex and non-continuousfunctions. On scaling-invariant functions, we show the existence of ahomogeneous Markov chain, as a consequence of natural invarianceproperties of CB-SARS (essentially scale-invariance and invariance tostrictly increasing transformation of the objective function). We thenderive sufficient conditions for \emph{global linear convergence} ofCB-SARS, expressed in terms of different stability conditions of thenormalised homogeneous Markov chain (irreducibility, positivity, Harrisrecurrence, geometric ergodicity) and thus define a general methodologyfor proving global linear convergence of CB-SARS algorithms onscaling-invariant functions. As a by-product we provide aconnexion between comparison-based adaptive stochasticalgorithms and Markov chain Monte Carlo algorithms.Comment: SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 201

Exploring the limits of the geometric copolymerization model

The geometric copolymerization model is a recently introduced statistical Markov chain model. Here, we investigate its practicality. First, several approaches to identify the optimal model parameters from observed copolymer fingerprints are evaluated using Monte Carlo simulated data. Directly optimizing the parameters is robust against noise but has impractically long running times. A compromise between robustness and running time is found by exploiting the relationship between monomer concentrations calculated by ordinary differential equations and the geometric model. Second, we investigate the applicability of the model to copolymerizations beyond living polymerization and show that the model is useful for copolymerizations involving termination and depropagation reactions

The Evolution of Dispersal in Random Environments and The Principle of Partial Control

McNamara and Dall (2011) identified novel relationships between the abundance of a species in different environments, the temporal properties of environmental change, and selection for or against dispersal. Here, the mathematics underlying these relationships in their two-environment model are investigated for arbitrary numbers of environments. The effect they described is quantified as the fitness-abundance covariance. The phase in the life cycle where the population is censused is crucial for the implications of the fitness-abundance covariance. These relationships are shown to connect to the population genetics literature on the Reduction Principle for the evolution of genetic systems and migration. Conditions that produce selection for increased unconditional dispersal are found to be new instances of departures from reduction described by the "Principle of Partial Control" proposed for the evolution of modifier genes. According to this principle, variation that only partially controls the processes that transform the transmitted information of organisms may be selected to increase these processes. Mathematical methods of Karlin, Friedland, and Elsner, Johnson, and Neumann, are central in generalizing the analysis. Analysis of the adaptive landscape of the model shows that the evolution of conditional dispersal is very sensitive to the spectrum of genetic variation the population is capable of producing, and suggests that empirical study of particular species will require an evaluation of its variational properties.Comment: Dedicated to the memory of Professor Michael Neumann, one of whose many elegant theorems provides for a result presented here. 28 pages, 1 table, 1 figur

Optimal recombination in genetic algorithms for combinatorial optimization problems: Part II

This paper surveys results on complexity of the optimal recombination problem (ORP), which consists in finding the best possible offspring as a result of a recombination operator in a genetic algorithm, given two parent solutions. In Part II, we consider the computational complexity of ORPs arising in genetic algorithms for problems on permutations: the Travelling Salesman Problem, the Shortest Hamilton Path Problem and the Makespan Minimization on Single Machine and some other related problems. The analysis indicates that the corresponding ORPs are NP-hard, but solvable by faster algorithms, compared to the problems they are derived from

Recursively accelerated multilevel aggregation for markov chains

Abstract. A recursive acceleration method is proposed for multiplicative multilevel aggregation algorithms that calculate the stationary probability vector of large, sparse, and irreducible Markov chains. Pairs of consecutive iterates at all branches and levels of a multigrid W cycle with simple, nonoverlapping aggregation are recombined to produce improved iterates at those levels. This is achieved by solving quadratic programming problems with inequality constraints: the linear combination of the two iterates is sought that has a minimal two-norm residual, under the constraint that all vector components are nonnegative. It is shown how the two-dimensional quadratic programming problems can be solved explicitly in an efficient way. The method is further enhanced by windowed top-level acceleration of the W cycles using the same constrained quadratic programming approach. Recursive acceleration is an attractive alternative to smoothing the restriction and interpolation operators, since the operator complexity is better controlled and the probabilistic interpretation of coarse-level operators is maintained on all levels. Numerical results are presented showing that the resulting recursively accelerated multilevel aggregation cycles for Markov chains, combined with top-level acceleration, converge significantly faster than W cycles and lead to close-to-linear computational complexity for challenging test problems