Effect of hill climbing in ga after reproduction for solving optimization problems

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Effect of Hill Climbing in GA After Reproduction For Solving Optimization Problems

Girdhar Gopal*, Rakesh Kumar, Naveen Kumar and Ishan JawaDepartment of Computer Science and Application

Kurukshetra University, Haryana 136119 India

Gopal et al. 2015. International J Ext Res. 3:79-86http://www.journalijer.com

*Corresponding author e-mail: [email protected]

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JOuRnal Of extensive ReseaRch e-Print ISSN: 2394-0301

IntroductionGenetic algorithms are adaptive algorithms proposed by John Holland (1975) and were described as adaptive heuristic search algorithms (Goldberg, 1989) based on the evolutionary ideas of natural selection and natural genetics by David Goldberg (1989). They are powerful optimization techniques that employ concepts of evolutionary biology to evolve optimal solutions to a given problem. Genetic algorithm works with a population of individu-als represented by chromosomes. Each chromosome is evaluated by its fitness value as computed by the objective function of the problem. The population undergoes transformation using three primary genetic operators – selection, crossover and mutation which form new generation of population. This process contin-ues to achieve the optimal solution. General structure of genetic algorithm is:

Procedure GA(fnx, n, r, m,ngen) //fnx is fitness function to evaluate individuals in population // n is the population size in each generation (say 10) // r is fraction of population generated by crossover (say 0.7) // m is the mutation rate (say 0.01) //ngen is total number of generations P := generate n individuals at random // initial generation is generated randomly nogen:=1 //denotes current generation while nogen <=ngen do {//Selection step: L:= Select(P,n,nogen) // n/2 individuals of P selected using any selection method.//Crossover step: S:= Crossover(L,n) // Generates n chromosomes using arithmetic crossover //Mutation step: Mutation(S,m) // Inversion of chromosomes with mutation rate m //Replacement step:

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AbstractGenetic algorithms (GA) are good to find optimum solutions for a broad class of problems. Hill Climbing is the oldest local search algorithm, which helps in solving optimization problems. One can combine these two traditional techniques to get the best from the mixture. In this paper, use of Hill climbing technique after reproduction operator in genetic algorithm is shown, it shows good improvements than Simple genetic algorithm. The experiments have been conducted using five dif-ferent benchmark functions and implementation is carried out using MATLAB. Results show the improvement over simple genetic algorithm.

Keywords: Genetic algorithm, hill climbing, crossover, mutation, reproduction.

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http://www.journalijer.comGopal et al. 2015. International J Ext Res. 3:79-86

P:=S pb(i):=min(fitfn(P)) // store best individual in population.i:=i+1 } best:=min(pb) //finds best individual in all generations

end proc

MethodsHybridization in Genetic AlgorithmsGenetic algorithm is a population based search technique. It searches through the state space by exploiting only the coding and objective function in each generation. Introducing problem spe-cific information in any of the genetic operators would form a hybrid algorithm. One technique of this mixing of problem spe-cific knowledge in GA is Memetic algorithm. Memetic algorithms are evolutionary algorithms which use local search with genetic algorithms to refine the individuals. They are inspired by Rich-ard Dawkin’s concept of meme, (Dawkins, 1976). Memetic algo-rithms are also called hybrid evolutionary algorithms, (Moscato and Cotta, 2003). Memetic algorithms can blend the functioning of genetic algorithm with other heuristic search techniques like hill climbing, tabu search etc.

Starting with the research contributed by Bosworth (Bosworth et al. 1972; Bethke, 1980; Brady, 1985), there is a long tradition of hybridizing Evolutionary Algorithms with other optimization methods such as Hill Climbing, Simulated Annealing, or Tabu Search (Sinha and Goldberg, 2003). A comprehensive review on this topic has been provided by Grosan and Abraham (2007). In-teresting research work directly focusing on Memetic Algorithms has been contributed, amongst many others, (Blesa et al. 2001; Bu-riol et al. 2004; Radcliffe and Surry, 1994; Digalakis and Margaritis, 2002; Krasnogor and Smith, 2005). Further early works on Genet-ic Algorithm hybridization ca be found, (Goldberg, 1989; Martina, 1989; Brown et al. 1989). Success of Genetic algorithm primarily depends on Initial population, (Martinez et al. 2004); Beam Search (BS) was integrated with genetic algorithm to seed the initial popu-lation. They started with BS and find the best product line design, and use it in initial population as a member, and then generate remaining N-1 population in a random fashion. Then proceed as with genetic algorithm termination when stopping condition met.

Hill ClimbingHill Climbing (HC) is one of the oldest and simplest local search and optimization algorithm for single objective functions f. The Hill Climbing procedures usually starts either from a random point in the search space or from a point chosen according to some problem-specific criterion. Then, in a loop, the currently known best individual p is modified (with the mutate( )operator) in order to produce one offspring pnew. If this new individual is better than its parent, it replaces it. Otherwise, it is discarded. This procedure is repeated until the termination criterion termination-cri-terion is met. The major problem of Hill Climbing is premature convergence, (Thomas, 2011).

HillClimbing(p)while (termination Criterion){pnew ←− mutate(p)if f(pnew.x) is Better Than f(p.x) p ← pnew}return p

Proposed Memetic AlgorithmGenetic algorithms are starting from a set of solutions, which gen-erated randomly. And then progress slowly towards optima. If we initialize the population with some heuristic search i.e. Hill Climb-ing, the new algorithm can result better in terms of optimum re-sult in less amount of time, also in terms of better optimum solu-tion. The proposed memetic algorithm is defined as:

Procedure GA(fnx, n, r, m, ngen) //fnx is fitness function to evaluate individuals in population // n is the population size in each generation (say 10) // r is fraction of population generated by crossover (say 0.7) // m is the mutation rate (say 0.01) //ngen is total number of generations P := generate n individuals at random // initial generation is generated randomly nogen:=1 //denotes current generation while nogen <=ngen do {//Selection step: L:= Select(P,n,nogen) // n/2 individuals of P selected using any selection method.//Crossover step: S:= Crossover(L,n) // Generates n chromosomes using arithmetic crossover //Mutation step: Mutation(S,m) // Inversion of chromosomes with mutation rate m //Apply Hill climbing on new chromosomes after the crossover and mutation. HillS = HillClimbing (S);//Replacement step: P:= HillS;pb(i):=min(fitfn(P)) // store best individual in population.i:=i+1 } best:=min(pb) //finds best individual in all generations

end proc

Results and Discussion(i) Experimental Set-up In this paper, four different functions are examined in order to compare performance of genetic algorithms and proposed mi-metic algorithm. Table 1 lists the five test functions – their names, type and their description.

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Function Name Type

F1 Sphere Function Unimodal

F2 Rosenbrock’s Function Unimodal

F3 Rastrigin’s Function Multimodal

F4 Schwefel’s Function Multimodal

Table 1. Description of Functions used in Experiments

First two functions are by De Jong and unimodal (only one opti-ma) functions, whereas, other two are multilodal (containing many local optimas, but only one global optima) functions. Sphere [F1] is simple quadratic parabola. It is smooth, unimodal, strongly con-vex, symmetric, (Digalakis and Margaritis, 2001; Salomon 1996).

Rosenbrock [F2] is considered to be difficult, because it has a very narrow ridge. The tip of the ridge is very sharp, and it runs around a parabola. The global optimum is inside a long, narrow

parabolic shaped flat valley, Digalakis and Margaritis, 2001; Salo-mon 1996).

Schwefel’s function (F4) is deceptive in that the global mini-mum is geometrically distant, over the parameter space, from the next best local minima. Therefore, the search algorithms are po-tentially prone to convergence in the wrong direction, Digalakis and Margaritis, 2001; Salomon 1996).

∑==

n

i ixxF1

2)(1

-5.12 ≤ x(i) ≤ 5.12global minimum: fn(x)=0, xi=0, i=1:n

2221

1 1 )1().(100 xxx ii

n

i i −+−∑ −

= +

-2.048<=x(i)<=2.048global minimum: f(x)=0; x(i)=1, i=1:n

F2(x) =


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F3(x) =

-5.12 ≤ x(i) ≤ 5.12global minimum: f(x)=0; x(i)=0, i=1:n

))..2cos(.10(.10 2

1 xx ii

n

in π−+∑ =

)sin(.)(41 xx i

n

i ixF ∑=−=

-500 ≤ x(i) ≤500global minimum: f(x)=-n•418.9829; x(i)=420.9687, i=1:n

The following parameters are used in this implementation: • Population size (N): 20, 50, 100• Number of generations (ngen) : 100 and 500• Selection method: Roulette Wheel Selection (RWS)

N 20 50 100

Gen = 100 SGA 0.012 0.0016 1.03E-08

MA 0.00883 3.72E-09 3.82E-08

Gen = 500 SGA 0.057 3.41E-06 5.13E-10

MA 3.42E-07 1.41E-08 5.37E-15

Table 2. Minimum values for F1

• Crossover Operator: Arithmetic Crossover (alpha = 0.35)• Mutation: uniform with mutation probability 0.1% • Algorithm ending criteria: Execution stops on reaching ngen

generations


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• Fitness Function: Objective value of function

(ii) Experimental ResultsAverage and minimum values of each function is recorded and examined for further analysis. Average and minimum values of each function is recorded and examined for further analysis.

Following two figures are the results of the simulation run in MATLAB for different generations showing minimum fitness of each generation.

N 2.00E+01 5.00E+01 100

Gen = 100 SGA 1.15E+00 2.90E-01 0.45

MA 2.90E-01 5.60E-03 0.0096

Gen = 500 SGA 0.56 0.45 0.024

MA 0.0011 0.0048 0.0005




N 2.00E+01 5.00E+01 100

Gen = 100 SGA 3.90E-02 3.89E-02 1.87E-04

MA 3.75E-06 5.32E-08 1.79E-08

Gen = 500 SGA 0.0049 1.98E-04 7.38E-07

MA 2.90E-07 1.39E-08 6.32E-10


N 2.00E+01 5.00E+01 100

Gen = 100 SGA -5.45E+02 -4.78E+02 -3.86E+02

MA -7.50E+02 -7.78E+02 -7.90E+02

Gen = 500 SGA -656 -8.34E+02 -8.10E+02

MA -8.12E+02 -7.99E+02 -8.16E+02



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ConclusionsWhile solving any optimization problem one must have to keep in mind that one solution for the entire optimization problems may never be found. It depends on the problem that one has to tune his solutions slightly according to the nature of the problem. Sometime local search is better also; no doubt global search has very few chances to get stuck in local optima. So the combination of these two schemes is also a balanced search method, which has speed of local search and accuracy of global search. When genetic algorithm generates new child for next generation then a tuning of them for their locality is a good promising achievement in the life cycle of genetic algorithm. This incorporation lead GA to the memetic algorithm. In this paper memetic algorithm shows better results in comparison to simple GA. So in future the exploitation of this hybridization may lead to better solutions.

Conflict of interestsThe authors declare that they have no conflict of interests.

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Article Information:Received: 12 February 2015 Accepted: 18 March 2015Online published: 25 March 2015

Cite this article as:G Gopal et al. 2015. Effect of hill climbing in GA after reproduction for solving optimization problems. International Journal of Extensive Research. Vol. 3: 79-86.

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