scholarly journals A Parameterised Complexity Analysis of Bi-level Optimisation with Evolutionary Algorithms

2016 ◽  
Vol 24 (1) ◽  
pp. 183-203 ◽  
Author(s):  
Dogan Corus ◽  
Per Kristian Lehre ◽  
Frank Neumann ◽  
Mojgan Pourhassan
Keyword(s):  
Evolutionary Algorithms ◽  
Evolutionary Algorithm ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Global Structure ◽  
Number Of Clusters ◽  
Structure Representation ◽  
Fixed Parameter ◽  
Travelling Salesperson Problem ◽  
Parameterised Complexity

Bi-level optimisation problems have gained increasing interest in the field of combinatorial optimisation in recent years. In this paper, we analyse the runtime of some evolutionary algorithms for bi-level optimisation problems. We examine two NP-hard problems, the generalised minimum spanning tree problem and the generalised travelling salesperson problem in the context of parameterised complexity. For the generalised minimum spanning tree problem, we analyse the two approaches presented by Hu and Raidl ( 2012 ) with respect to the number of clusters that distinguish each other by the chosen representation of possible solutions. Our results show that a (1+1) evolutionary algorithm working with the spanning nodes representation is not a fixed-parameter evolutionary algorithm for the problem, whereas the problem can be solved in fixed-parameter time with the global structure representation. We present hard instances for each approach and show that the two approaches are highly complementary by proving that they solve each other’s hard instances very efficiently. For the generalised travelling salesperson problem, we analyse the problem with respect to the number of clusters in the problem instance. Our results show that a (1+1) evolutionary algorithm working with the global structure representation is a fixed-parameter evolutionary algorithm for the problem.

scholarly journals A dandelion-encoded evolutionary algorithm for the delay-constrained capacitated minimum spanning tree problem

2009 ◽  
Vol 32 (1) ◽  
pp. 154-158 ◽  
Author(s):  
Angel M. Pérez-Bellido ◽  
Sancho Salcedo-Sanz ◽  
Emilio G. Ortiz-Garcı´a ◽  
Antonio Portilla-Figueras ◽  
Maurizio Naldi
Keyword(s):  
Evolutionary Algorithm ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Capacitated Minimum Spanning Tree ◽  
Minimum Spanning Tree Problem

Application of optimization tasks in cluster analysis

10.12737/7483 ◽  
2014 ◽  
Vol 8 (7) ◽  
pp. 0-0
Author(s):  
Олег Сдвижков ◽  
Oleg Sdvizhkov
Keyword(s):  
Cluster Analysis ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Clustering Algorithms ◽  
Clustering Methods ◽  
Number Of Clusters ◽  
Homogeneous Groups ◽  
Finite Set ◽  
Minimum Number

Cluster analysis [3] is a relatively new branch of mathematics that studies the methods partitioning a set of objects, given a finite set of attributes into homogeneous groups (clusters). Cluster analysis is widely used in psychology, sociology, economics (market segmentation), and many other areas in which there is a problem of classification of objects according to their characteristics. Clustering methods implemented in a package STATISTICA [1] and SPSS [2], they return the partitioning into clusters, clustering and dispersion statistics dendrogram of hierarchical clustering algorithms. MS Excel Macros for main clustering methods and application examples are given in the monograph [5]. One of the central problems of cluster analysis is to define some criteria for the number of clusters, we denote this number by K, into which separated are a given set of objects. There are several dozen approaches [4] to determine the number K. In particular, according to [6], the number of clusters K - minimum number which satisfies where - the minimum value of total dispersion for partitioning into K clusters, N - number of objects. Among the clusters automatically causes the consistent application of abnormal clusters [4]. In 2010, proposed and experimentally validated was a method for obtaining the number of K by applying the density function [4]. The article offers two simple approaches to determining K, where each cluster has at least two objects. In the first number K is determined by the shortest Hamiltonian cycles in the second - through the minimum spanning tree. The examples of clustering with detailed step by step solutions and graphic illustrations are suggested. Shown is the use of macro VBA Excel, which returns the minimum spanning tree to the problems of clustering. The article contains a macro code, with commentaries to the main unit.

scholarly journals ETEA: A Euclidean Minimum Spanning Tree-Based Evolutionary Algorithm for Multi-Objective Optimization

2014 ◽  
Vol 22 (2) ◽  
pp. 189-230 ◽  
Author(s):  
Miqing Li ◽  
Shengxiang Yang ◽  
Jinhua Zheng ◽  
Xiaohui Liu
Keyword(s):  
Evolutionary Algorithm ◽  
Spanning Tree ◽  
Euclidean Distance ◽  
Minimum Spanning Tree ◽  
Optimization Problems ◽  
Multi Objective Optimization ◽  
Position Information ◽  
Multi Objective ◽  
Environmental Selection ◽  
Fitness Value

The Euclidean minimum spanning tree (EMST), widely used in a variety of domains, is a minimum spanning tree of a set of points in space where the edge weight between each pair of points is their Euclidean distance. Since the generation of an EMST is entirely determined by the Euclidean distance between solutions (points), the properties of EMSTs have a close relation with the distribution and position information of solutions. This paper explores the properties of EMSTs and proposes an EMST-based evolutionary algorithm (ETEA) to solve multi-objective optimization problems (MOPs). Unlike most EMO algorithms that focus on the Pareto dominance relation, the proposed algorithm mainly considers distance-based measures to evaluate and compare individuals during the evolutionary search. Specifically, in ETEA, four strategies are introduced: (1) An EMST-based crowding distance (ETCD) is presented to estimate the density of individuals in the population; (2) A distance comparison approach incorporating ETCD is used to assign the fitness value for individuals; (3) A fitness adjustment technique is designed to avoid the partial overcrowding in environmental selection; (4) Three diversity indicators—the minimum edge, degree, and ETCD—with regard to EMSTs are applied to determine the survival of individuals in archive truncation. From a series of extensive experiments on 32 test instances with different characteristics, ETEA is found to be competitive against five state-of-the-art algorithms and its predecessor in providing a good balance among convergence, uniformity, and spread.

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