MIC/MAEB 2017
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Subdividing Labeling Genetic Algorithm: A New Method for Solving Continuous Nonlinear Optimization Problems Majid Esmaelian1, Madjid Tavana2, Francisco J. Santos-Arteaga3, Masoumeh Vali4 1
Department of Management University of Isfahan, Isfahan, Iran
[email protected] 2
Business Systems and Analytics Department Distinguished Chair of Business Analytics La Salle University, Philadelphia, PA 19141, USA
[email protected] 2
Business Information Systems Department Faculty of Business Administration and Economics University of Paderborn, D-33098 Paderborn, Germany 3
School of Economics and Management Free University of Bolzano, Bolzano, Italy
[email protected] 4
Young Researchers and Elite Club Khorasgan Branch, Islamic Azad University, Isfahan, Iran
[email protected] Abstract In most global optimization problems, finding a global optimum point in the whole multi-dimensional search space implies a high computational burden. We present a new approach called subdividing labeling genetic algorithm (SLGA) for continuous nonlinear optimization problems. SLGA applies mutation and crossover operators on a subdivided search space where an integer label is defined on a polytope built on a n dimensional space. After calculating the fitness of each point composing the polytope, SLGA implements a mutation operator to generate offspring and computes an integer label for the population of the polytope. Then, after completely labeling the polytope, a crossover operator is implemented so as to approach the optimum point by reducing the search space. In this regard, new population is generated by subdividing the search space and further implementing the mutation operator. SLGA has been used to optimize the De Jong's functions, as well as nonlinear constrained and unconstrained problems with discrete, continuous and mixed variables. It has also been compared with other well-known algorithms. Experimental results show that the SLGA method has good performance and reduces the number of generations within the solution space, which enhances its convergence capability.
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Introduction
We propose a new approach to optimize a simple-bounded continuous function, denoted by f (x 1, x 2 ,..., x n ) , in which ai x i bi (i=1,2,...,n) , for some constants ai and bi . The aim of the approach is to find the optimum point based on a modified version of a genetic algorithm. To be consistent with the notation of genetic-based methods, we refer to f as the fitness function throughout the paper. Figure 1 below provides an overview of the proposed subdividing labeling genetic algorithm. 1. Similarly to standard genetic algorithms, the proposed approach starts from an initial population P , which contains the vertices of the polytope built using the boundaries of x i . 2. An initial optimum point S is selected from this population using an elitism mechanism. For example, if the goal is to minimize the fitness function, then the optimum point is the one leading Barcelona, July 4-7, 2017
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MIC/MAEB 2017
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to the lowest fitness value. Then, a mutation operator is implemented to generate offspring from the initial population P . The main aim of this operator is to approach the optimum point by reducing the search space. The resulting n dimensional points are labeled so as to obtain a completely labeled polytope. A crossover operation is performed between the optimum point S and the completely labeled polytope. In particular, the crossover operation defines a weighted mean between the optimum point S and the vertices and center point of the adjacent sides of the completely labeled polytope. Note that the size of the new population is equal to four and consists of the point S and the three offsprings generated via crossover. The optimum n dimensional offspring S ' is selected from the population using an elitism mechanism, i.e. according to the value of the fitness function f . The optimum offspring obtained from the new population is denoted by S ' . If the algorithm meets the precision requirements, then the process ends and S ' is the optimum point. However, if the algorithm does not meet the precision requirements, a new population is generated by subdividing the search space and implementing the mutation operator. The algorithm proceeds by labeling the new polytope and repeating the above crossover operation between the optimum point S ' and the new completely labeled polytope so as to obtain a new optimum point.
Figure 1: Proposed SLGA algorithm flowchart
Barcelona, July 4-7, 2017
