scholarly journals Lower Bounds for Non-Elitist Evolutionary Algorithms via Negative Multiplicative Drift

2020 ◽  
pp. 1-25
Author(s):  
Benjamin Doerr
Keyword(s):  
Evolutionary Algorithms ◽  
Lower Bounds ◽  
Population Based ◽  
Mutation Rates ◽  
Reproduction Rate ◽  
Random Mutation ◽  
Search Spaces ◽  
Discrete Search ◽  
Negative Drift ◽  
Special Case

A decent number of lower bounds for non-elitist population-based evolutionary algorithms has been shown by now. Most of them are technically demanding due to the (hard to avoid) use of negative drift theorems — general results which translate an expected movement away from the target into a high hitting time. We propose a simple negative drift theorem for multiplicative drift scenarios and show that it can simplify existing analyses. We discuss in more detail Lehre's (PPSN 2010) negative drift in populations method, one of the most general tools to prove lower bounds on the runtime of non-elitist mutation-based evolutionary algorithms for discrete search spaces. Together with other arguments, we obtain an alternative and simpler proof of this result, which also strengthens and simplifies this method. In particular, now only three of the five technical conditions of the previous result have to be verified. The lower bounds we obtain are explicit instead of only asymptotic. This allows to compute concrete lower bounds for concrete algorithms, but also enables us to show that super-polynomial runtimes appear already when the reproduction rate is only a [Formula: see text] factor below the threshold. For the special case of algorithms using standard bit mutation with a random mutation rate (called uniform mixing in the language of hyper-heuristics), we prove the result stated by Dang and Lehre (PPSN 2016) and extend it to mutation rates other than [Formula: see text], which includes the heavytailed mutation operator proposed by Doerr, Le, Makhmara, and Nguyen (GECCO 2017). We finally use our method and a novel domination argument to show an exponential lower bound for the runtime of the mutation-only simple genetic algorithm on ONEMAX for arbitrary population size.

Runtime Analysis of the (μ + 1) EA on Simple Pseudo-Boolean Functions

2006 ◽  
Vol 14 (1) ◽  
pp. 65-86 ◽  
Author(s):  
Carsten Witt
Keyword(s):  
Lower Bounds ◽  
Boolean Functions ◽  
Success Probability ◽  
Population Based ◽  
Upper And Lower Bounds ◽  
Rigorous Analysis ◽  
Search Spaces ◽  
Discrete Search ◽  
Family Trees ◽  
Theoretical Developments

Although Evolutionary Algorithms (EAs) have been successfully applied to optimization in discrete search spaces, theoretical developments remain weak, in particular for population-based EAs. This paper presents a first rigorous analysis of the (μ + 1) EA on pseudo-Boolean functions. Using three well-known example functions fromthe analysis of the (1 + 1) EA, we derive bounds on the expected runtime and success probability. For two of these functions, upper and lower bounds on the expected runtime are tight, and on all three functions, the (μ + 1) EA is never more efficient than the (1 + 1) EA. Moreover, all lower bounds growwith μ. On a more complicated function, however, a small increase of μ provably decreases the expected runtime drastically. This paper develops a newproof technique that bounds the runtime of the (μ + 1) EA. It investigates the stochastic process for creating family trees of individuals; the depth of these trees is bounded. Thereby, the progress of the population towards the optimum is captured. This new technique is general enough to be applied to other population-based EAs.

scholarly journals Bounding the Size and Probability of Epidemics on Networks

2008 ◽  
Vol 45 (2) ◽  
pp. 498-512 ◽  
Author(s):  
Joel C. Miller
Keyword(s):  
Infectious Disease ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Marginal Probability ◽  
Transmission Properties ◽  
Disease Spreading ◽  
Special Case

We consider an infectious disease spreading along the edges of a network which may have significant clustering. The individuals in the population have heterogeneous infectiousness and/or susceptibility. We define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility. For a given distribution of out-transmissibility, we find the conditions which give the upper (or lower) bounds on the size and probability of an epidemic, under weak assumptions on the transmission properties, but very general assumptions on the network. We find similar bounds for a given distribution of in-transmissibility (the marginal probability of being infected by a neighbor). We also find conditions giving global upper bounds on the size and probability. The distributions leading to these bounds are network independent. In the special case of networks with high girth (locally tree-like), we are able to prove stronger results. In general, the probability and size of epidemics are maximal when the population is homogeneous and minimal when the variance of in- or out-transmissibility is maximal.

scholarly journals Blood Coagulation Algorithm: A Novel Bio-Inspired Meta-Heuristic Algorithm for Global Optimization

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3011
Author(s):  
Drishti Yadav
Keyword(s):  
Blood Coagulation ◽  
Heuristic Algorithms ◽  
Cooperative Behavior ◽  
Real Life ◽  
Search Space ◽  
Population Based ◽  
Test Analysis ◽  
Search Spaces ◽  
Consistent Performance ◽  
The Comparative Study

This paper introduces a novel population-based bio-inspired meta-heuristic optimization algorithm, called Blood Coagulation Algorithm (BCA). BCA derives inspiration from the process of blood coagulation in the human body. The underlying concepts and ideas behind the proposed algorithm are the cooperative behavior of thrombocytes and their intelligent strategy of clot formation. These behaviors are modeled and utilized to underscore intensification and diversification in a given search space. A comparison with various state-of-the-art meta-heuristic algorithms over a test suite of 23 renowned benchmark functions reflects the efficiency of BCA. An extensive investigation is conducted to analyze the performance, convergence behavior and computational complexity of BCA. The comparative study and statistical test analysis demonstrate that BCA offers very competitive and statistically significant results compared to other eminent meta-heuristic algorithms. Experimental results also show the consistent performance of BCA in high dimensional search spaces. Furthermore, we demonstrate the applicability of BCA on real-world applications by solving several real-life engineering problems.

scholarly journals On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

10.37236/969 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Jörn Quistorff
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Finite Sets ◽  
Open Problems ◽  
Latin Rectangle ◽  
Mixed Codes ◽  
Special Case

Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

Lower bounds for cutting planes proofs with small coefficients

1997 ◽  
Vol 62 (3) ◽  
pp. 708-728 ◽  
Author(s):  
Maria Bonet ◽  
Toniann Pitassi ◽  
Ran Raz
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Proof System ◽  
Small Weight ◽  
Deduction Rule ◽  
Special Case ◽  
Exponential Lower Bound

AbstractWe consider small-weight Cutting Planes (CP*) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution.We also prove the following two theorems: (1) Tree-like CP* proofs cannot polynomially simulate non-tree-like CP* proofs. (2) Tree-like CP* proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other.Our proofs also work for some generalizations of the CP* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

Parallel Evolutionary Computation in R

2013 ◽  
pp. 105-129 ◽  
Author(s):  
Cedric Gondro ◽  
Paul Kwan
Keyword(s):  
Evolutionary Algorithms ◽  
Evolutionary Computation ◽  
Genome Wide Association Study ◽  
Optimization Methods ◽  
Large Field ◽  
Search Spaces ◽  
Parallel Evolutionary Algorithms ◽  
Genome Wide ◽  
Statistical Programming ◽  
Candidate Regions

Evolutionary Computation (EC) is a branch of Artificial Intelligence which encompasses heuristic optimization methods loosely based on biological evolutionary processes. These methods are efficient in finding optimal or near-optimal solutions in large, complex non-linear search spaces. While evolutionary algorithms (EAs) are comparatively slow in comparison to deterministic or sampling approaches, they are also inherently parallelizable. As technology shifts towards multicore and cloud computing, this overhead becomes less relevant, provided a parallel framework is used. In this chapter the authors discuss how to implement and run parallel evolutionary algorithms in the popular statistical programming language R. R has become the de facto language for statistical programming and it is widely used in biostatistics and bioinformatics due to the availability of thousands of packages to manipulate and analyze data. It is also extremely easy to parallelize routines within R, which makes it a perfect environment for evolutionary algorithms. EC is a large field of research, and many different algorithms have been proposed. While there is no single silver bullet that can handle all classes of problems, an algorithm that is extremely simple, efficient, and with good generalization properties is Differential Evolution (DE). Herein the authors discuss step-by-step how to implement DE in R and how to parallelize it. They then illustrate with a toy genome-wide association study (GWAS) how to identify candidate regions associated with a quantitative trait of interest.

scholarly journals On the number of excursion sets of planar Gaussian fields

2020 ◽  
Vol 178 (3-4) ◽  
pp. 655-698
Author(s):  
Dmitry Beliaev ◽  
Michael McAuley ◽  
Stephen Muirhead
Keyword(s):  
Plane Wave ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Gaussian Fields ◽  
Important Special Case ◽  
Level Densities ◽  
Excursion Sets ◽  
Nodal Set ◽  
Different Types ◽  
Special Case

Abstract The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.

On the development of cat swarm metaheuristic using distributed learning strategies and the applications

2019 ◽  
Vol 12 (2) ◽  
pp. 224-244 ◽  
Author(s):  
Usha Manasi Mohapatra ◽  
Babita Majhi ◽  
Alok Kumar Jagadev
Keyword(s):  
Nonlinear Systems ◽  
Evolutionary Algorithms ◽  
Learning Strategies ◽  
Input Data ◽  
Distributed Learning ◽  
Identification Problem ◽  
Population Based ◽  
Communication Overhead ◽  
Swarm Optimization ◽  
Content Type

Purpose The purpose of this paper is to propose distributed learning-based three different metaheuristic algorithms for the identification of nonlinear systems. The proposed algorithms are experimented in this study to address problems for which input data are available at different geographic locations. In addition, the models are tested for nonlinear systems with different noise conditions. In a nutshell, the suggested model aims to handle voluminous data with low communication overhead compared to traditional centralized processing methodologies. Design/methodology/approach Population-based evolutionary algorithms such as genetic algorithm (GA), particle swarm optimization (PSO) and cat swarm optimization (CSO) are implemented in a distributed form to address the system identification problem having distributed input data. Out of different distributed approaches mentioned in the literature, the study has considered incremental and diffusion strategies. Findings Performances of the proposed distributed learning-based algorithms are compared for different noise conditions. The experimental results indicate that CSO performs better compared to GA and PSO at all noise strengths with respect to accuracy and error convergence rate, but incremental CSO is slightly superior to diffusion CSO. Originality/value This paper employs evolutionary algorithms using distributed learning strategies and applies these algorithms for the identification of unknown systems. Very few existing studies have been reported in which these distributed learning strategies are experimented for the parameter estimation task.

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