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Chaotic harmony search optimizer for solving numerical integration.

1. Introduction

Numerical integration plays a very important and critical role in various research areas. A formula for the integrand may be known, but it may be difficult or impossible to find an anti-derivative which is an elementary function. It may be possible to find an anti-derivative symbolically, but it may be easier to compute a numerical approximation than to compute the anti-derivative. There are some traditional methods such as Newton method, Gauss method, Romberg method, and Simpson's method (Mi,1991), but they have many limitations like complexity in calculations, low precision, and their convergence is not guaranteed for higher order (Qu, 2010).

The Harmony Search (HS) algorithm originally came from the analogy between music improvisation and optimization process (Geem, 2001). This algorithm has been successfully applied to various discrete optimization problems such as traveling salesperson problem (Geem, 2001), tour routing (Geem, 2005), music composition (Geem, 2007), and water network design (Geem, 2006). A new method is developed based on Harmony Search algorithm combined with Chaos theory to solve numerical integration. The main idea behind this new method is to make adaptive optimization of the segmentations which are represented by harmony vectors generated randomly in the integral interval.

This paper is structured as following: section 2 is made for harmony search approach, section 3 is devoted to Chaos, the proposed algorithm is illustrated in section 4, experiments and simulation results are shown in section 5, and finally in section 6 conclusion is presented.

2. Harmony Search Algorithm

The HS algorithm was originally developed by Geem et al. in 2001, and is based on natural musical performance processes that occur when a musician searches for a better state of harmony, such as during jazz improvisation. Jazz improvisation seeks to find musically pleasing harmony (a perfect state) as determined by an aesthetic standard, just as the optimization process seeks to find a global solution (a perfect state) as determined by an objective function. The pitch of each musical instrument determines the aesthetic quality, just as the objective function value is determined by the set of values assigned to each design variable (Lee, 2005). The HS algorithm optimization procedure which is shown below consists of the following five steps.

Step 1: Problem and algorithm parameter initialization

The optimization problem is specified as follows:

Minimize f(x) subject to [X.sub.j] [member of] [X.sub.j] = 1,2, ... N, where f(x) is an objective function; x is the set of each decision variable [x.sub.j] N is the number of decision variables, [X.sub.j] is the set of the possible range of values for each decision variable, that is [x.sup.min.sub.j] and [x.sup.max.sub.j] are the lower and upper boundaries of the jth decision parameter respectively. The HS algorithm parameters are also specified in this step. These are the harmony memory size (HMS), or the number of

solution vectors in the harmony memory; harmony memory considering rate (HMCR); pitch adjusting rate (PAR); bandwidth distance (bw); and the number of improvisations (NI), or stopping criterion. The harmony memory (HM) is a memory location where all the solution vectors (sets of decision variables) are stored.

Step 2: Harmony memory initialization and evaluation

The HM matrix is randomly generated as shown in Eq. (1)

[x.sup.0.sub.i,j] = [x.sup.min.sub.j] + [r.sub.j] x (x.sup.max.sub.j] - [x.sup.min.sub.j]) i = 1,2, ..., HMS; j = 1,2, ..., N (1)

and [r.sub.j] [member of] [0,1] is a uniformly distributed random number generated new for each value of j. Solution vectors in HM are analyzed, and their objective function values are calculated.

Step 3: New harmony improvisation

In this step, a new harmony vector is generated based on three rules. They are memory consideration, pitch adjustment, and random selection. The value of a design variable can be selected from the values stored in HM with a probability of harmony memory considering rate (HMCR). It can be further adjusted by moving to a neighbor value of a selected value from the HM with a probability of pitch adjusting rate (PAR). Or, it can be selected randomly from the set of all candidate values without considering the stored values in HM, with the probability of (1--HMCR).

Step 4: Harmony memory update

The new better harmony vector is included in the HM and the worst harmony is excluded.

Step 5: Termination criterion check

The HS algorithm is terminated when the termination criterion (e.g. maximum number of improvisations) has been met. Otherwise, steps 3 and 4 are repeated.

3. Chaos

In recent years, the theories and applications of nonlinear dynamics, especially of chaos, have drawn more and more attention in many fields. One is chaos controlling, and synchronization. Another field is the potential applications of chaos in various disciplines including optimization (Yang, 2007).

Chaos is a deterministic, random-like process found in nonlinear, dynamical system, which is non-period, non-converging and bounded. Moreover, it has a very sensitive dependence upon its initial condition and parameter. The nature of chaos is apparently random and unpredictable and it also possesses an element of regularity. Mathematically, chaos is randomness of a simple deterministic dynamical system and chaotic system may be considered as sources of randomness (Alatas, 2010). A chaotic map is a discrete-time dynamical system

[x.sub.k+1] = f ([x.sub.k]), 0 < [x.sub.k] < 1, k = 0,1,2, ... (2)

running in the chaotic state. The chaotic sequence {[x.sub.k] : k = 0,1,2,.} can be used as spread-spectrum sequence and as a random number sequence.

Some new searching algorithms called Chaos Optimization Algorithms (COAs) use the properties of chaos like ergodicity as in Li (1998) and Zhang (1999). Recently Chaos is extended to various optimization areas, for example Multi-Objective Optimization (El-Santawy, 2011) because it can more easily escape from local minima than other stochastic optimization algorithms. The use of chaotic sequences in HS can be helpful to improve the global convergence, and to prevent sticking on a local solution than the classical HS algorithm which uses fixed values for HMCR, PAR and bw. In the chaotic HS algorithm, when a random number is needed by the classical HS algorithm, it is generated by iterating one step of the chosen chaotic map that has been started from a random initial condition at the first iteration of the HS (Alatas, 2010).

4. Chaotic harmony Search for Solving Numerical Integrals

Suppose that segmentation T splits an integral interval [a,b] into n-subintervals:

[[x.sub.0],[x.sub.1]],[[x.sub.1],[x.sub.2]],.,[[x.sub.k-1],[x.sub.k]],[[x.sub.n-1],[x.sub.n]], where [X.sub.j] < [x.sub.j+1] for j = 1,2, ..., n-1; [x.sub.0] = a, and [x.sub.n] = b, also define [DELTA][x.sub.k] = [x.sub.k] - [x.sub.k-1] for k = 1,2,.,n. Using this notation, the integral f(x) in [a,b] can be approximated as in Eq. (3) (Qu, 2010):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The steps of the Chaotic Harmony Search algorithm for calculating numerical value of definite Integral (CHSINT) are as follows:

Step 1: setting the HS algorithm parameters { HMS,HMCR,NI}.

Step 2: initializing HM by iterating the selected chaotic maps until if reaches the HMS. This process produces HMS segmentations randomly in [a,b]. Determining the fitness of each harmony vector as in Eq. (3) is done after.

Step 3: generating new harmony improvisations based on the three updating rules as illustrated in section 2. In this algorithm PAR and bw values have not been fixed in HS and they have been modified by the selected chaotic maps as follows

PAR(t+1) = f (PAR(t)), 0 < PAR(t) < 1, (4) t = 1,2, ...

bw(t+1) = f (bw(t)), 0 < bw(t) < 1, (5) t = 1,2, ...

Step 4: updating Harmony memory after evaluating all harmony vectors in Eq. (3).

Step 5: checking to ensure the termination criterion has been met. Otherwise, steps 3 and 4 are repeated.

5. Experiments and Results

Several experiments have been done to verify the validity of the proposed algorithm. The initial parameters set at HMS = 40, HMCR = 0.9, and NI = 50. The selected chaotic map for all experiments is the Logistic map, whose equation is given in Eq. (6). It has been brought to the attention of scientists by Robert May (May, 1976).

[z.sub.k+1] = [mu][z.sub.k] (1- [Z.sub.k]) (6)

Obviously, [z.sub.k] [member of] [0,1] under the conditions that the initial [z.sub.0] [member of] [0,1], where k is the iteration number and p = 4. The programs are realized in Matlab 7.0. The integral values of functions 1/(1+x) in [0,2], [e.sup.x] in [0,2], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in [0,1] are selected for experiments. The results of CHSINT algorithm are conducted from 30 independent runs for each integrand, while the results of other techniques are found in Burden (2001) and Wang (2004). As shown in Table 1 the results of CHSINT algorithm are superior compared to the results of Trapezoidal method, Simpson's method, and Fourth order Newton-Cotes method.

6. Conclusion

In this paper, a novel algorithm based on Harmony search and Chaos for calculating the numerical value of definite integrals is presented. This algorithm has the ability to overcome the shortage that the segmentation points are uniform in traditional methods. The simulation examples of numerical integration validated the algorithm as effective and enforceable. Experimental results show that the algorithm can converge to the best solution, and it has high accuracy which makes it easily usable in engineering.

References

Alatas, B. (2010),"Chaotic harmony search algorithms", Applied Mathematics and Computation, 216: 2687-2699.

Burden, R.L., Faires, J.D. (2001), Numerical Analysis, 7th edition. Brooks /Cole, Thomson Learning, Inc.

El-Santawy, M. F. and Ahmed, A. N. (2011), "The chaotic e-constraint approach", Journal of American Science, 7(7): 470-472.

Geem, Z. W., Kim, J. H. and Loganathan, G. V. (2001), "A New Heuristic Optimization Algorithm: Harmony Search", Simulation, 76(2): 60-68.

Geem, Z. W., Tseng, C. -L. and Park, Y. (2005), "Harmony Search for Generalized Orienteering Problem: Best Touring in China", Lecture Notes in Computer Science, 3612: 741-750.

Geem, Z. W. (2006), "Optimal Cost Design of Water Distribution Networks using Harmony Search", Engineering Optimization, 38(3): 259-280.

Geem, Z. W. and Choi, J. Y. (2007), "Music Composition Using Harmony Search Algorithm", Lecture Notes in Computer Science, 4448: 593-600.

Li, B. and Jiang, W. (1998),"Optimization of complex functions by chaos search", International Journal of Cybernetics and Systems, 29(4): 409-419.

Lee, K.S., Geem, Z.W., Lee, S.-H. and Bae, K.W. (2005), "The harmony search heuristic algorithm for discrete structural optimization", Engineering Optimization, 37(7): 663-684.

May, R.M. (1976), "Simple mathematical models with very complicated dynamics", Nature, 26: 459-467.

Mi, X. (1991), Value mathematics and calculates, Fudan University Publishing House, Shanghai, China.

Qu, L. and He, D. (2010), "Solving Numerical Integration by Particle Swarm Optimization", Communications in Computer and Information Science, 106(4): 228-235.

Wang, X.H., He, Y.G. and Zeng, Z.Z. (2004), "Numerical Integration Study Based on triangle Basis Neural Network Algorithm", Journal of Electronics & Information Technology, 26(3): 394-399.

Yang, D., Li, G. and Cheng, G. (2007), "On the efficiency of chaos optimization algorithms for global optimization", Chaos, Solitons and Fractals, 34: 1366-1375.

Zhang, T., Wang, H. and Wang, Z. (1999),"Mutative scale chaos optimization algorithm and its application", Control and Decision 14(3): 285-288.

Mohamed F. El-Santawy (1)

lost_zola@yahoo.com

A. N. Ahmed (2)

drhadi@cu.edu.eg

Ramadan A. Zean El-Dean (3)

rzeanedean@yahoo.com

(1,3) Department of Operations Research, Institute of Statistical Studies and Research (ISSR), Cairo University, Egypt

(2) Department of Mathematical Statistics, Institute of Statistical Studies and Research (ISSR), Cairo University, Egypt
Table 1. Integral values of functions

       f (x)          1/(1+x)   [e.sup.x]   [MATHEMATICAL
                                            EXPRESSION
                                            NOT
                                            REPRODUCIBLE
                                            IN ASCII]

CHSINT                1.0986    6.3891      0.74682
Trapezoidal method    1.3333    8.3891      0.74621
Simpson's method      1.1111    6.4207      0.74683
Newton-Cotes method   1.0993    6.3892      0.74683
Accurate value        1.0986    6.3891      0.74682
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Author:El-Santawy, Mohamed F.; Ahmed, A.N.; Dean, Ramadan A. Zean El-
Publication:Computing and Information Systems
Geographic Code:1USA
Date:May 1, 2012
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