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 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.

Some Properties of Divisor Methods for Legislative Apportionment and Proportional Representation

1982 ◽  
Vol 76 (3) ◽  
pp. 575-584 ◽  
Author(s):  
Cyril Carter
Keyword(s):  
Probability Distribution ◽  
Lower Bounds ◽  
Proportional Representation ◽  
Upper And Lower Bounds ◽  
Rigorous Analysis ◽  
Political Choice ◽  
Divisor Methods ◽  
Divisor Method

Some rather elegant properties of linear divisor methods are derived and used to establish upper and lower bounds on the possible variation between apportionment and exact quota entitlement. A probability distribution is derived for this variation, and it is shown that the probability of the variation exceeding one seat is very small with the major fractions linear divisor method.A less rigorous analysis of the nonlinear equal proportions method shows that in practice it is very similar to the major fractions method, but with a very slight bias in favor of small parties (or states). It is concluded that there is no “best” apportionment method, but a knowledge of the properties of the various methods enables a political choice of the most appropriate method.

scholarly journals Anytime Recursive Best-First Search for Bounding Marginal MAP

2019 ◽  
Vol 33 ◽  
pp. 7924-7932
Author(s):  
Radu Marinescu ◽  
Akihiro Kishimoto ◽  
Adi Botea ◽  
Rina Dechter ◽  
Alexander Ihler
Keyword(s):  
Lower Bounds ◽  
Graphical Models ◽  
Memory Management ◽  
Empirical Evaluation ◽  
Optimal Solution ◽  
Difficult Problem ◽  
Upper And Lower Bounds ◽  
Search Spaces ◽  
Best First Search ◽  
Problem Instances

Marginal MAP is a difficult mixed inference task for graphical models. Existing state-of-the-art solvers for this task are based on a hybrid best-first and depth-first search scheme that allows them to compute upper and lower bounds on the optimal solution value in an anytime fashion. These methods however are memory intensive schemes (via the best-first component) and do not have an efficient memory management mechanism. For this reason, they are often less effective in practice, especially on difficult problem instances with very large search spaces. In this paper, we introduce a new recursive best-first search based bounding scheme that operates efficiently within limited memory and computes anytime upper and lower bounds that improve over time. An empirical evaluation demonstrates the effectiveness of our proposed approach against current solvers.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

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