An Evaluation of Methods for Estimating the Number of Local Optima in Combinatorial Optimization Problems

2013 ◽  
Vol 21 (4) ◽  
pp. 625-658 ◽  
Author(s):  
Leticia Hernando ◽  
Alexander Mendiburu ◽  
Jose A. Lozano
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Search Space ◽  
Combinatorial Optimization Problem ◽  
Local Search Algorithms ◽  
Neighborhood Structure ◽  
Local Optima ◽  
Combinatorial Optimization Problems ◽  
Evaluation Of Methods

The solution of many combinatorial optimization problems is carried out by metaheuristics, which generally make use of local search algorithms. These algorithms use some kind of neighborhood structure over the search space. The performance of the algorithms strongly depends on the properties that the neighborhood imposes on the search space. One of these properties is the number of local optima. Given an instance of a combinatorial optimization problem and a neighborhood, the estimation of the number of local optima can help not only to measure the complexity of the instance, but also to choose the most convenient neighborhood to solve it. In this paper we review and evaluate several methods to estimate the number of local optima in combinatorial optimization problems. The methods reviewed not only come from the combinatorial optimization literature, but also from the statistical literature. A thorough evaluation in synthetic as well as real problems is given. We conclude by providing recommendations of methods for several scenarios.

scholarly journals Anatomy of the Attraction Basins: Breaking with the Intuition

2019 ◽  
Vol 27 (3) ◽  
pp. 435-466 ◽  
Author(s):  
Leticia Hernando ◽  
Alexander Mendiburu ◽  
Jose A. Lozano
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Local Search Algorithms ◽  
Neighborhood Structure ◽  
Natural Landscape ◽  
Local Optima ◽  
Attraction Basin ◽  
Combinatorial Optimization Problems ◽  
New Findings ◽  
Attraction Basins

Solving combinatorial optimization problems efficiently requires the development of algorithms that consider the specific properties of the problems. In this sense, local search algorithms are designed over a neighborhood structure that partially accounts for these properties. Considering a neighborhood, the space is usually interpreted as a natural landscape, with valleys and mountains. Under this perception, it is commonly believed that, if maximizing, the solutions located in the slopes of the same mountain belong to the same attraction basin, with the peaks of the mountains being the local optima. Unfortunately, this is a widespread erroneous visualization of a combinatorial landscape. Thus, our aim is to clarify this aspect, providing a detailed analysis of, first, the existence of plateaus where the local optima are involved, and second, the properties that define the topology of the attraction basins, picturing a reliable visualization of the landscapes. Some of the features explored in this article have never been examined before. Hence, new findings about the structure of the attraction basins are shown. The study is focused on instances of permutation-based combinatorial optimization problems considering the 2-exchange and the insert neighborhoods. As a consequence of this work, we break away from the extended belief about the anatomy of attraction basins.

scholarly journals A Methodology to Find the Elementary Landscape Decomposition of Combinatorial Optimization Problems

2011 ◽  
Vol 19 (4) ◽  
pp. 597-637 ◽  
Author(s):  
Francisco Chicano ◽  
L. Darrell Whitley ◽  
Enrique Alba
Keyword(s):  
Combinatorial Optimization ◽  
Objective Function ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Quadratic Assignment Problem ◽  
Combinatorial Optimization Problem ◽  
Quadratic Assignment ◽  
Neighborhood Structure ◽  
Combinatorial Optimization Problems ◽  
Elementary Landscape

A small number of combinatorial optimization problems have search spaces that correspond to elementary landscapes, where the objective function f is an eigenfunction of the Laplacian that describes the neighborhood structure of the search space. Many problems are not elementary; however, the objective function of a combinatorial optimization problem can always be expressed as a superposition of multiple elementary landscapes if the underlying neighborhood used is symmetric. This paper presents theoretical results that provide the foundation for algebraic methods that can be used to decompose the objective function of an arbitrary combinatorial optimization problem into a sum of subfunctions, where each subfunction is an elementary landscape. Many steps of this process can be automated, and indeed a software tool could be developed that assists the researcher in finding a landscape decomposition. This methodology is then used to show that the subset sum problem is a superposition of two elementary landscapes, and to show that the quadratic assignment problem is a superposition of three elementary landscapes.

Space Contraction and Partition Algorithm for Combinatorial Optimization

2011 ◽  
Vol 421 ◽  
pp. 559-563
Author(s):  
Yong Chao Gao ◽  
Li Mei Liu ◽  
Heng Qian ◽  
Ding Wang
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Solution Space ◽  
Search Space ◽  
Small Scale ◽  
Optimal Solutions ◽  
Solution Quality ◽  
Intelligent Algorithms ◽  
Combinatorial Optimization Problems

The scale and complexity of search space are important factors deciding the solving difficulty of an optimization problem. The information of solution space may lead searching to optimal solutions. Based on this, an algorithm for combinatorial optimization is proposed. This algorithm makes use of the good solutions found by intelligent algorithms, contracts the search space and partitions it into one or several optimal regions by backbones of combinatorial optimization solutions. And optimization of small-scale problems is carried out in optimal regions. Statistical analysis is not necessary before or through the solving process in this algorithm, and solution information is used to estimate the landscape of search space, which enhances the speed of solving and solution quality. The algorithm breaks a new path for solving combinatorial optimization problems, and the results of experiments also testify its efficiency.

Matched Field Processing Based on the Simulated Annealing

2013 ◽  
Vol 651 ◽  
pp. 879-884
Author(s):  
Qi Wang ◽  
Ying Min Wang ◽  
Yan Ni Gou
Keyword(s):  
Global Optimization ◽  
Simulated Annealing ◽  
Combinatorial Optimization ◽  
Mediterranean Sea ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Combinatorial Optimization Problem ◽  
Matched Field Processing ◽  
Combinatorial Optimization Problems ◽  
Passive Localization

The matched field processing (MFP) for localization usually needs to match all the replica fields in the observation sea with the received fields, and then find the maximum peaks in the matched results, so how to find the maximum in the results effectively and quickly is a problem. As known the classical simulated annealing (CSA) which has the global optimization capability is used widely for combinatorial optimization problems. For passive localization the position of the source can be recognized as a combinatorial optimization problem about range and depth, so a new matched field processing based on CSA is proposed. In order to evaluate the performance of this method, the normal mode was used to calculate the replica field. Finally the algorithm was evaluated by the dataset in the Mediterranean Sea in 1994. Comparing to the conventional matched field passive localization (CMFP), it can be conclude that the new one can localize optimum peak successfully where the output power of CMFP is maximum, meanwhile it is faster than CMFP.

scholarly journals An Interactive Regret-Based Genetic Algorithm for Solving Multi-Objective Combinatorial Optimization Problems

2020 ◽  
Vol 34 (03) ◽  
pp. 2335-2342
Author(s):  
Nawal Benabbou ◽  
Cassandre Leroy ◽  
Thibaut Lust
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Computation Time ◽  
Preference Elicitation ◽  
Combinatorial Optimization Problem ◽  
New Approach ◽  
Multi Objective ◽  
Combinatorial Optimization Problems

We propose a new approach consisting in combining genetic algorithms and regret-based incremental preference elicitation for solving multi-objective combinatorial optimization problems with unknown preferences. For the purpose of elicitation, we assume that the decision maker's preferences can be represented by a parameterized scalarizing function but the parameters are initially not known. Instead, the parameter imprecision is progressively reduced by asking preference queries to the decision maker during the search to help identify the best solutions within a population. Our algorithm, called RIGA, can be applied to any multi-objective combinatorial optimization problem provided that the scalarizing function is linear in its parameters and that a (near-)optimal solution can be efficiently determined when preferences are known. Moreover, RIGA runs in polynomial time while asking no more than a polynomial number of queries. For the multi-objective traveling salesman problem, we provide numerical results showing its practical efficiency in terms of number of queries, computation time and gap to optimality.

Study on Free Search for Combinatorial Optimization Problem

2013 ◽  
Vol 411-414 ◽  
pp. 1904-1910
Author(s):  
Kai Zhong Jiang ◽  
Tian Bo Wang ◽  
Zhong Tuan Zheng ◽  
Yu Zhou
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problem ◽  
Feasible Solution ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Search Process ◽  
Standard Data ◽  
Combinatorial Optimization Problems ◽  
Free Search

An algorithm based on free search is proposed for the combinatorial optimization problems. In this algorithm, a feasible solution is converted into a full permutation of all the elements and a transformation of one solution into another solution can be interpreted the transformation of one permutation into another permutation. Then, the algorithm is combined with intersection elimination. The discrete free search algorithm greatly improves the convergence rate of the search process and enhances the quality of the results. The experiment results on TSP standard data show that the performance of the proposed algorithm is increased by about 2.7% than that of the genetic algorithm.

scholarly journals Selected Soft Computing Algorithms for Job Shop Problem (JSP)

2021 ◽  
Vol 10 (9) ◽  
pp. 125-131
Author(s):  
Adedeji Oluyinka Titilayo ◽  
Alade Oluwaseun Modupe ◽  
Makinde Bukola Oyeladun ◽  
OYELEYE Taye E
Keyword(s):  
Combinatorial Optimization ◽  
Computer Science ◽  
Soft Computing ◽  
Operations Research ◽  
Optimization Problem ◽  
Job Shop ◽  
Optimization Problems ◽  
Combinatorial Optimization Problems ◽  
Flower Pollination

Job Shop Problem (JSP) is an optimization problem in computer science and operations research in which jobs are assigned to resources at particular times. Each operation has a specific machine that it needs to be processed on and only one operation in a job can be processed at a given time. This problem is one of the best known combinatorial optimization problems. The aim of this project is to adapt Bat, Bee, Firefly, and Flower pollination algorithms, implement and evaluate the developed algorithms for solving Job Shop Problem.

Advances of Applying Metaheuristics to Data Mining Techniques

2015 ◽  
pp. 75-103 ◽  
Author(s):  
Simon Fong ◽  
Jinyan Li ◽  
Xueyuan Gong ◽  
Athanasios V. Vasilakos
Keyword(s):  
Data Mining ◽  
Combinatorial Optimization ◽  
Job Scheduling ◽  
Optimization Problems ◽  
Local Optima ◽  
Processing Resource ◽  
Combinatorial Optimization Problems ◽  
Data Mining Algorithms ◽  
Biological Entities ◽  
Mining Algorithms

Metaheuristics have lately gained popularity among researchers. Their underlying designs are inspired by biological entities and their behaviors, e.g. schools of fish, colonies of insects, and other land animals etc. They have been used successfully in optimization applications ranging from financial modeling, image processing, resource allocations, job scheduling to bioinformatics. In particular, metaheuristics have been proven in many combinatorial optimization problems. So that it is not necessary to attempt all possible candidate solutions to a problem via exhaustive enumeration and evaluation which is computationally intractable. The aim of this paper is to highlight some recent research related to metaheuristics and to discuss how they can enhance the efficacy of data mining algorithms. An upmost challenge in Data Mining is combinatorial optimization that, often lead to performance degradation and scalability issues. Two case studies are presented, where metaheuristics improve the accuracy of classification and clustering by avoiding local optima.

Website link Structure Optimization

2016 ◽  
pp. 222-251 ◽  
Author(s):  
Harpreet Singh ◽  
Parminder Kaur
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Structure Optimization ◽  
Optimization Models ◽  
Graph Structure ◽  
Link Structure ◽  
Combinatorial Optimization Problems ◽  
Web Graph ◽  
The Web

The structure of a website can be represented in the form of a graph where nodes represent pages and edges represent hyperlinks among those pages. The behaviour of website users changes continuously and hence the link structure of a website should be modified frequently. The problem of optimally rearranging the link structure of a website is known as Website Structure Optimization problem. It falls in the category of combinatorial optimization problems. Many methods have been proposed and developed by the researchers to optimize the web graph structure of a website. In this chapter taxonomy of the website link structure optimization models is presented. The formulation and explanation of the working of the models have also been provided so that the readers could easily understand the methodology used by the models.

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