Exact Solutions to the Traveling Salesperson Problem by a Population-Based Evolutionary Algorithm

2009 ◽  
pp. 145-155 ◽  
Author(s):  
Madeleine Theile
Keyword(s):  
Exact Solutions ◽  
Evolutionary Algorithm ◽  
Population Based ◽  
Traveling Salesperson Problem

Parameterized Runtime Analyses of Evolutionary Algorithms for the Planar Euclidean Traveling Salesperson Problem

2014 ◽  
Vol 22 (4) ◽  
pp. 595-628 ◽  
Author(s):  
Andrew M. Sutton ◽  
Frank Neumann ◽  
Samadhi Nallaperuma
Keyword(s):  
Evolutionary Algorithms ◽  
Evolutionary Algorithm ◽  
Upper Bound ◽  
Domain Knowledge ◽  
Population Based ◽  
Theoretical Understanding ◽  
Traveling Salesperson Problem ◽  
Fitness Evaluation ◽  
Expected Time ◽  
Mutation Strategy

Parameterized runtime analysis seeks to understand the influence of problem structure on algorithmic runtime. In this paper, we contribute to the theoretical understanding of evolutionary algorithms and carry out a parameterized analysis of evolutionary algorithms for the Euclidean traveling salesperson problem (Euclidean TSP). We investigate the structural properties in TSP instances that influence the optimization process of evolutionary algorithms and use this information to bound their runtime. We analyze the runtime in dependence of the number of inner points k. In the first part of the paper, we study a [Formula: see text] EA in a strictly black box setting and show that it can solve the Euclidean TSP in expected time [Formula: see text] where A is a function of the minimum angle [Formula: see text] between any three points. Based on insights provided by the analysis, we improve this upper bound by introducing a mixed mutation strategy that incorporates both 2-opt moves and permutation jumps. This strategy improves the upper bound to [Formula: see text]. In the second part of the paper, we use the information gained in the analysis to incorporate domain knowledge to design two fixed-parameter tractable (FPT) evolutionary algorithms for the planar Euclidean TSP. We first develop a [Formula: see text] EA based on an analysis by M. Theile, 2009, ”Exact solutions to the traveling salesperson problem by a population-based evolutionary algorithm,” Lecture notes in computer science, Vol. 5482 (pp. 145–155), that solves the TSP with k inner points in [Formula: see text] generations with probability [Formula: see text]. We then design a [Formula: see text] EA that incorporates a dynamic programming step into the fitness evaluation. We prove that a variant of this evolutionary algorithm using 2-opt mutation solves the problem after [Formula: see text] steps in expectation with a cost of [Formula: see text] for each fitness evaluation.

scholarly journals Parameterized Analysis of Multiobjective Evolutionary Algorithms and the Weighted Vertex Cover Problem

2019 ◽  
Vol 27 (4) ◽  
pp. 559-575
Author(s):  
Mojgan Pourhassan ◽  
Feng Shi ◽  
Frank Neumann
Keyword(s):  
Evolutionary Algorithm ◽  
Complexity Analysis ◽  
Minimum Weight ◽  
Vertex Cover ◽  
Optimal Solution ◽  
Population Based ◽  
Mutation Operator ◽  
Vertex Cover Problem ◽  
Weighted Vertex ◽  
Cover Problem

Evolutionary multiobjective optimization for the classical vertex cover problem has been analysed in Kratsch and Neumann ( 2013 ) in the context of parameterized complexity analysis. This article extends the analysis to the weighted vertex cover problem in which integer weights are assigned to the vertices and the goal is to find a vertex cover of minimum weight. Using an alternative mutation operator introduced in Kratsch and Neumann ( 2013 ), we provide a fixed parameter evolutionary algorithm with respect to [Formula: see text], the cost of an optimal solution for the problem. Moreover, we present a multiobjective evolutionary algorithm with standard mutation operator that keeps the population size in a polynomial order by means of a proper diversity mechanism, and therefore, manages to find a 2-approximation in expected polynomial time. We also introduce a population-based evolutionary algorithm which finds a [Formula: see text]-approximation in expected time [Formula: see text].

Imperialist Competitive Algorithms with Perturbed Moves for Global Optimization

2013 ◽  
Vol 284-287 ◽  
pp. 3135-3139 ◽  
Author(s):  
Jun Lin Lin ◽  
Chun Wei Cho ◽  
Hung Chjh Chuan
Keyword(s):  
Global Optimization ◽  
Evolutionary Algorithm ◽  
Imperialist Competitive Algorithm ◽  
Population Based ◽  
Experimental Results ◽  
Local Optimum ◽  
Competitive Algorithm ◽  
Competitive Algorithms

Imperialist Competitive Algorithm (ICA) is a new population-based evolutionary algorithm. Previous works have shown that ICA converges quickly but often to a local optimum. To overcome this problem, this work proposed two modifications to ICA: perturbed assimilation move and boundary bouncing. The proposed modifications were applied to ICA and tested using six well-known benchmark functions with 30 dimensions. The experimental results indicate that these two modifications significantly improve the performance of ICA on all six benchmark functions.

A HYBRID EVOLUTIONARY ALGORITHM FOR MULTIPLE-DESTINATIONS ROUTING PROBLEM

2004 ◽  
Vol 04 (04) ◽  
pp. 337-353 ◽  
Author(s):  
SALAH AL-SHARHAN ◽  
FAWAZ AL-ANZI
Keyword(s):  
Evolutionary Algorithm ◽  
Incremental Learning ◽  
Heuristic Algorithms ◽  
Learning Algorithm ◽  
Population Based ◽  
Multicast Tree ◽  
Routing Problem ◽  
Hybrid Evolutionary Algorithm ◽  
Np Complete ◽  
Qos Metrics

This paper presents a hybrid evolutionary algorithm for constrained multiple destinations routing problem. The problem can be formulated as minimising tree cost under several constraints or QoS metrics. Computing such constrained multicast tree has been proven to be NP-complete. The proposed hybrid algorithm is based on a population based incremental learning algorithm and a constrained distance network heuristic (or CKMB) algorithm. In the proposed algorithm, CKMB is utilised as a decoding scheme. Experimental results show that, in most cases, the proposed algorithm yields better solutions than other heuristic algorithms proposed in the literature including the best known one BSMA.

DIFFERENTIAL EVOLUTION FOR SOLVING MULTIOBJECTIVE OPTIMIZATION PROBLEMS

2004 ◽  
Vol 21 (02) ◽  
pp. 225-240 ◽  
Author(s):  
RUHUL SARKER ◽  
HUSSEIN A. ABBASS
Keyword(s):  
Differential Evolution ◽  
Evolutionary Algorithm ◽  
Optimization Problems ◽  
Pareto Frontier ◽  
Population Based ◽  
Standard Test ◽  
Evolutionary Strategies ◽  
Test Problems ◽  
Strength Pareto Evolutionary Algorithm ◽  
Multiobjective Optimization Problems

The use of evolutionary strategies (ESs) to solve problems with multiple objectives [known as vector optimization problems (VOPs)] has attracted much attention recently. Being population-based approaches, ESs offer a means to find a set of Pareto-optimal solutions in a single run. Differential evolution (DE) is an ES that was developed to handle optimization problems over continuous domains. The objective of this paper is to introduce a novel Pareto-frontier differential evolution (PDE) algorithm to solve VOPs. The solutions provided by the proposed algorithm for two standard test problems, outperform the "strength Pareto evolutionary algorithm", one of the state-of-the-art evolutionary algorithm for solving VOPs.

Constrained Optimization of a Rotor-Bearing System Based on an Evolutionary Algorithm

2012 ◽  
Author(s):  
Donghua Wang ◽  
Wanyou Li ◽  
Zhigang Liu ◽  
Zhansheng Liu
Keyword(s):  
Evolutionary Algorithm ◽  
Optimal Solution ◽  
Population Based ◽  
Rotor System ◽  
System Configuration ◽  
Critical Speeds ◽  
Rotor Systems ◽  
Elite Strategy ◽  
Practical Design ◽  
Bearing System

How to modify a rotor system configuration to adjust some critical speeds distribution for safety and achieve the least change of system configuration at the same time, is a focus in the rotordynamics field. An existing method is introduced and its disadvantages are analyzed. To overcome these difficulties smoothly, a more robust mathematical model for optimal design of critical speeds distribution is presented, with more constraints considered. And a Population-based Evolutionary Algorithm with Elite Strategy (PEAES) is developed to solve the proposed optimization model. The results of study on a rotor system in different cases show that the proposed method can find the optimal solution and may be applicable in the practical design process of rotor systems.

Solution of “Hard” Knapsack Instances Using Quantum Inspired Evolutionary Algorithm

2014 ◽  
Vol 5 (1) ◽  
pp. 52-68 ◽  
Author(s):  
C. Patvardhan ◽  
Sulabh Bansal ◽  
Anand Srivastav
Keyword(s):  
Evolutionary Algorithm ◽  
Exact Algorithm ◽  
Population Based ◽  
Exact Algorithms ◽  
Programming Technique ◽  
Worst Case ◽  
Problem Instances ◽  
Population Based Search ◽  
Made In

Knapsack Problem (KP) is a popular combinatorial optimization problem having application in many technical and economic areas. Several attempts have been made in past to solve the problem. Various exact and non-exact approaches exist to solve KP. Exact algorithms for KP are based on either branch and bound or dynamic programming technique. Heuristics exist which solve KP non-exactly in lesser time. Heuristic approaches do not provide any guarantee regarding the quality of solution whereas exact approaches have high worst case complexities. Quantum-inspired Evolutionary Algorithm (QEA) is a subclass of Evolutionary Algorithm, a naturally inspired population based search technique. QEA uses concepts of quantum computing. An engineered Quantum-inspired Evolutionary Algorithm (QEA-E), an improved version of QEA, is presented which quickly solves extremely large spanner problem instances (e.g. 290,000 items) that are very difficult for the state of the art exact algorithm as well as the original QEA.

scholarly journals A Probabilistic Analysis of a Simplified Biogeography-Based Optimization Algorithm

2011 ◽  
Vol 19 (2) ◽  
pp. 167-188 ◽  
Author(s):  
Dan Simon
Keyword(s):  
Probability Theory ◽  
Evolutionary Algorithm ◽  
Population Size ◽  
Optimization Algorithm ◽  
Probabilistic Analysis ◽  
Population Based ◽  
Approximate Analysis ◽  
Expected Number ◽  
Expected Values ◽  
Biological Organisms

Biogeography-based optimization (BBO) is a population-based evolutionary algorithm (EA) that is based on the mathematics of biogeography. Biogeography is the study of the geographical distribution of biological organisms. We present a simplified version of BBO and perform an approximate analysis of the BBO population using probability theory. Our analysis provides approximate values for the expected number of generations before the population's best solution improves, and the expected amount of improvement. These expected values are functions of the population size. We quantify three behaviors as the population size increases: first, we see that the best solution in the initial randomly generated population improves; second, we see that the expected number of generations before improvement increases; and third, we see that the expected amount of improvement decreases.

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