Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 620325 | 16 pages | https://doi.org/10.1155/2014/620325

Quantum-Inspired Evolutionary Algorithm for Continuous Space Optimization Based on Multiple Chains Encoding Method of Quantum Bits

Academic Editor: Jingjing Zhou
Received03 Apr 2014
Accepted14 Jun 2014
Published23 Jul 2014

Abstract

This study proposes a novel quantum evolutionary algorithm called four-chain quantum-inspired evolutionary algorithm (FCQIEA) based on the four gene chains encoding method. In FCQIEA, a chromosome comprises four gene chains to expand the search space effectively and promote the evolutionary rate. Different parameters, including rotational angle and mutation probability, have been analyzed for better optimization. Performance comparison with other quantum-inspired evolutionary algorithms (QIEAs), evolutionary algorithms, and different chains of QIEA demonstrates the effectiveness and efficiency of FCQIEA.

1. Introduction

Quantum computation is based on the principal concepts of the quantum theory [1, 2]. Numerous researchers have devoted increasing interests to quantum computation, a novel interdisciplinary field that covers quantum mechanics and information science. Quantum-inspired evolutionary algorithms (QIEA) focus on generating new evolutionary algorithms (EA) in accordance with some concepts and principles of quantum computation, such as standing waves [3]. In QIEA, the quantum bits (qubits) that encode chromosomes are used and quantum gate (Q-gate) has been proposed as a tool that can be used to update the chromosome to achieve evolutionary search. The binary observation QIEA (bQIEA), based on a probability optimization algorithm according to quantum computation concept and theory, was initially proposed in 2002 by Han and Kim to solve combinatorial optimization problems [4]. Since then, variants of the bQIEA have been developed, including bQIEA with different operators, such as crossover and mutation operators [58], bQIEA with a novel update method for Q-gates [912], and hybrid bQIEA that focuses on the interactions between bQIEA and common genetic algorithm (CGA), immune algorithms, and particle swarm optimization (PSO) [1231]. The combination of quantum computation and genetic algorithm has been widely studied [16, 3243]. QIEA has also been proposed as a method that can be applied to solve some combinatorial optimization problems [4466], such as traveling salesman problem [32], knapsack problem [67], and filter design [68], as well as design optimizations of electromagnetic devices [69] and analog test point selection [70]. QIEA has numerous advantages, such as the use of a small population size, fast convergence, and an excellent global-search capability compared with conventional EAs.

However, several studies have demonstrated that bQIEA is not suitable for numerical optimization problems [16, 71], such as function extremum optimization. Several problems have caused bQIEA to have slow converging speed and prematurity. For instance, frequent binary encoding and quantum chromosome decoding cause randomness and blindness. The magnitude and the direction of the rotational angle of quantum rotation gates cannot be determined during optimization. To solve global numerical optimization problems with continuous variables, studies have proposed a real-observation QIEA (rQIEA) [72, 73], which is characterized by the modified Q-bit representation to represent a Q-bit individual, the use of real observation to generate real-valued solutions from Q-bit individuals, and the modified Q-gate to guide the individuals toward better solutions. S. Li and P. Li [74] further proposed a convenient method to determine the direction of the rotational angle; in particular, a real qubits encoding method can be used to avoid binary coding and the double-chains quantum evolutionary algorithm, which is characterized by each qubit that is regarded as two coordinate genes. Each chromosome contains two gene chains, and each gene chain represents an optimization result. P. Li and S. Li [75] studied the appropriate rotational angle size between two rotational angles to obtain the best optimization effects and proposed the three-chain quantum evolutionary algorithm, which is characterized by bloch coordinates as the gene chains, such that each chromosome has three gene chains.

The optimized effects of three-chain quantum evolutionary algorithm is more efficient than the double-chains quantum evolutionary algorithm in solving continuous space optimization problems because the three-chain quantum evolutionary algorithm can accelerate the evolution speed and expand the search space of QIEA by increasing the number of gene chains. The current study presents a novel QIEA called four-chain quantum-inspired evolutionary algorithm (FCQIEA) based on expanded chains encoding to improve the QIEA performance further. FCQIEA integrates the characteristics of the double-chains encoding method and the three-chain encoding method to describe each qubit as four paratactic genes. Our results demonstrate that FCQIEA is more effective and feasible based on the optimization experiments that involve typical function extremum. We also compared our results with common quantum-inspired evolutionary algorithm (CQIEA), double-chains quantum-inspired evolutionary algorithm (DCQIEA), and three-chain quantum-inspired evolutionary algorithm (TCQIEA). In addition, we compared the performance of the proposed FCQIEA with other QIEAs and other EAs.

2. The Principle of FCQIEA

2.1. General Description of Continuous Optimization

In general, a continuous space optimization problem can be described as follows:

In (1), -dimensional continuous space optimization solutions are regarded as points or vectors. and are the lower and the upper bounds for the design variable , respectively. Hence, restrained conditions are bounded as closed set in -dimension continuous space, and all points in are regarded as approximate solutions.

2.2. Expanded Encoding Method for Quantum Chromosome

In quantum computation, the basic unit of information is described by a qubit, which can be expressed as follows: where and satisfy the following normalization condition:

A pair of complex numbers and in (2) and (3) is called qubit probability amplitude. Therefore, the qubit can also be described by using the probability amplitude as . According to the nature of the probability amplitude, a qubit can be expressed as in Figure 1.

Thus, and ; then, qubit can be expressed as follows:

Qubits are directly regarded as genes on a chromosome; thus, we can use (4) as the encoding method. The principle of the double-chains encoding method [43] is applied and described as follows: where , , and is a random number from 0 to 1; and ; describes the population size; is the number of qubits.

We can consider points [0, 0.5] and [0, −0.05] as the center of two semicircles, such as in the Chinese Taiji (Figure 2).

Figure 2 shows that the vector of qubit intersects the semicircle at point , which can be connected to point . We can draw the vertical line segment from point to line segment . We can infer that angle forms a semicircle and angle . Thus, the length of the line segment . Similarly, . Thus, we can obtain the following equations:

Similarly, we obtain the following:

Equations (6) and (7) show that each vector in a qubit can be a hypotenuse in a right triangle so that it can be separated into two vectors, in which the sum of squares is equal to the square of the vector. Hence, we can determine vector as follows:

Equation (8) is consistent with the following normalization condition and thus is equivalent to

We can also substitute (4) in (10) as follows:

To describe the quantum dynamics behavior objectively, comprehensively, and unambiguously, we can use a new angle (), which is called “supporting role,” to replace and obtain vector as follows:

Equation (11) also satisfies the normalization condition; hence, (11) is correct. In fact, (11) also corresponds to the three-chain encoding method [75]:

Likewise, we can obtain vector by adding the “supporting role” to form the four-chain encoding method as follows:

We can obtain four optimal solutions, which are expressed below:

We define , , , and as solutions , , , and , respectively. The encoded chromosomal structure is shown in Figure 3.

Figure 3 shows that each chromosome can be separated into four gene chains, and each gene chain can have infinite sets of qubits that greatly expand the number of the global optimal solution and significantly increase the probability to obtain the global optimal solution.

2.3. Solution Space Transformation

In the quantum evolution progress, all qubits have limited values within −1 to 1. Thus, we need to transform all qubit values from unit space to space of the continuous optimization problem (1) by using linear transformation. Each gene value corresponds to an optimization variable in the solution space. If the qubit on chromosome is , then the corresponding variables in the solution space are computed as follows: where and . Thus, each chromosome maps out four approximate solutions of the optimization problem.

2.4. Quantum Chromosome Update

Considering that quantum chromosomes are present in the colony and we can obtain approximate solutions by solution space transformation, we can then compute the fitness of these approximate solutions and define the solution with the maximum fitness as the current optimum solution in the quantum evolution progress. The chromosome corresponds to the current optimum solution called the optimum chromosome. By computing the fitness, we can obtain both optimum solution and optimum chromosome and subsequently update the colony by using the quantum rotation gate to obtain the optimal solution. In this updated process, the new optimum chromosome can be produced such that the colony can likely evolve. The present study proposes the quantum rotation gate to update the individual qubit as follows:

These equations show that we can obtain the phase rotation of , , and from . , , and are important factors that directly influence the convergence speed and the efficiency of quantum evolution algorithm. In general, a query table is constructed [4] to determine the size and the direction of , , and . In this way, all of the possible instances are considered when making a decision, but it is very complicated to lower the algorithm efficiency. To determine the rule that considers the size and the direction of the rotational angle by using a simple method, we can deduce the following conclusions.

Theorem 1. Suppose is the th qubit in the current optimum chromosome of the four gene chains encoding method; then, is the th qubit in the th chromosome, where and .

Let (1)The direction of is determined by the following rules. If , then ; if , then the direction of may be positive or negative at random.(2)The direction of is determined by the following rules. If , then ; if , then the direction of may be positive or negative at random.(3)The direction of is determined by the following rules. If , then ; if , then the direction of may be positive or negative at random.

Proof. Consider the following equations:

To determine the direction of , is expressed in the form of a trigonometric function as follows: We can obtain from , , , and .(1)If , then If , then or .If , then at least one angle is zero; and can have arbitrary values. Therefore, the direction of is arbitrary. If , then or ; therefore, both positive and negative directions have the same result. Thus, the direction of is arbitrary when .(2)If , then If , then or ; both positive and negative directions have the same result. Therefore, the direction of is arbitrary.(3)If , then If , then or ; both positive and negative directions have the same result. Therefore, the direction of is arbitrary.

The magnitudes of , , and have an important effect on the convergence speed. A decreased efficiency is observed in the optimization process when the magnitudes of , , and are very small. By contrast, a premature convergence likely occurs in the optimization process when the magnitudes of , , and are very high.

The values from to are recommended for the magnitudes of , , and , although they depend on the problems [75].

Based on our analysis, a simplified query table (Table 1) is constructed to determine the direction of , , and .


0
0
0

Let chromosome be . Based on Table 1, the update progress of is described by a pseudocode as shown in Pseudocode 1.

Procedure update ( )
Begin
  
   While ( ) Do
      Begin
       
       Determine , , and with the query direction (Table 1) and obtain as follows:
       
    End
    
End

2.5. Mutation Operation

Quantum nongate is applied to exchange the probability amplitudes to avoid local optimal solution in a certain qubit as follows:

Such influence as expressed in (18) can be considered as the phase mutation of a qubit, in which is mutated to . In this case, a quantum nongate is proposed to mutate the quantum as follows:

The quantum nongate rotation can enhance population diversity. The mutation operation does not compare with the current optimum chromosome. The rotation magnitudes are usually great, which are , , and .

Let the qubits in the chromosome be , where and ; is the mutation probability. Based on our analysis, the mutation progress is described by another pseudocode as shown in Pseudocode 2.

Procedure Mutate ( )
Begin
 If is not the current optimum chromosome,
  Begin
  
  While ( )
  Begin
  
  Generate Random Number rnd in range 0 to 1
  If rnd < , then obtain from
  End
    
 End
End

3. Algorithm Description

The structure of FCQIEA is shown by the pseudocode under the section of the procedure for FCQIEA (see Pseudocode 3).

Begin
(1) Initialize the colony .
(2) Construct by solution space transformation.
(3) Obtain the optimum solution of the current among and the current  optimum chromosome among
  by evaluating .
(4) Store into and into .
While (not termination-condition)
 Begin
(5) Calculate by updating and mutating the states of .
   Determine by solution space transformation.
(6) Obtain the optimum solution of the current among and the current optimum chromosome among
    by evaluating .
(7) If
;
 Otherwise
;
 End
  End
End
(8) The procedure of FCQIEA can be summarized as follows:
   Step  1. Initialize the population. Let the current generation ; generate an initial
population , which has individual qubits. Set the magnitude of the rotational angle ,
, and , respectively. Set as the mutation and as the maximum generation.
   Step  2. Transform the solution space. Four approximate solutions in each chromosome are transformed from the unit space
to the solution space of the continuous optimization problem (1); thus, the set of approximate solution
can be obtained.
   Step  3. Compute the fitness. By computing the fitness of approximate solutions, obtain the best solution in the
current solution and the best chromosome in the current chromosome. Store as the global optimum solution
and store as the global optimum chromosome .
   Step  4. Set . Update and mutate . Calculate the new population .
   Step  5. Transform the solution space again and obtain a set of approximate solution .
   Step  6. By computing the fitness of , determine the current optimum solution and the current optimum.
chromosome If , then update the current optimum solution ; at the same time,
update the current optimum chromosome to avoid population degradation. Otherwise, let
and so that the algorithm approaches the global optimum solution.
   Step  7. If the algorithm does not converge and if , then go back to Step 4 until the algorithm
becomes convergent or until .

4. Convergence Analysis

Considering that the population is , the quantum chromosome is defined as follows:

Lemma 2. The sequence of iteration in FCQIEA is a finite Markov chain.

Proof. Given that the value of is continuous, the state space of the population is theoretically infinite, but is finite in accuracy in the progress of computation. Considering that the value of is and represents the magnitude of the state space, the population becomes finite. The chromosome is then updated and mutated to show the relationship between and ; thus, the iteration sequence of is a finite Markov chain (end).

Lemma 3. In the Markov chain sequence, which is generated by the iteration of FCQIEA, the sequence of the best fitness in the population is not continuously decreasing. Thus, the equation is always when .

Proof. By the retention strategy of storing the best chromosome in FCQIEA, the population is not degenerated in each population generation. The population is expressed as when .

Lemma 4. FCQIEA is converged with probability 1.

Proof. Assume that qubits are present in each chromosome in a population with chromosomes, which can be described as point in state space. The state space of the population is defined as when the accuracy of is finite and the value is . Consider as the best individual in from the population; then, is defined as the global optimum solution set and as the global best fitness. Suppose and represents the state after times iteration (); then, the transition probability can be calculated in the random progress after further iteration from the state to the state. (1)If and , by Lemma 3 and , then .(2)If and , by Lemma 3 and , then .
Considering that is the probability with the state in population and , the probability of with the state of can be derived using the Markov chain as follows:
The probability of can be further derived as follows: Thus,

5. Experimental Result

We tested the FCQIEA performance by using different parameters to achieve the best performance for FCQIEA. Afterward, the performance of FCQIEA was compared with CQIEA, DCQIEA, and TCQIEA, as well as with bQIEA and rQIEA. The FCQIEA performance was also compared with the standard genetic algorithm (GA), particle swarm optimization (PSO), and artificial bee colony (ABC).

5.1. Parameter Analysis

In this section, different parameters, which include rotational angle and mutation probability, were analyzed by conducting a simulation test with two application examples of extremum optimizing complex functions (Goldstein-Price function and Shubert function) to achieve the best performance of FCQIEA.

(1) The Goldstein-Price function is expressed as follows: where and . The minimum local point is expressed as follows:

, , , and .

The global minimum is 3. Optimum results <3.005 can be the result of the algorithm convergence. Goldstein-Price function is shown in Figure 4.

(2) The Shubert function is expressed as follows: where . This function is a typical multipeak function, in which the maximum extremum value point is that corresponds to the global optimum value of 10. When the optimum results reach 9.995, the algorithm is considered as convergent. Shubert function is shown in Figure 5.

The range of , , and was studied by applying Goldstein-Price and Shubert functions. The population size, the maximum number of epochs, and the mutation probability are set at 10, 100, and 0, respectively.

The optimum result can be expressed as shown in Figures 6 and 7 when .

Figures 6 and 7 show that the optimization results are optimal when .

The relationship between , , and was also investigated. The population size, the maximum number of epochs, and the mutation probability are set at 10, 100, and 0, respectively.

Let The optimization results are shown in Figures 8 and 9 when .

Figures 8 and 9 show that the value of , , and considers the following six cases when we obtain the optimum result:(1), ;(2), ;(3), ;(4), ;(5), ;(6), .

Based on our analysis, the optimization result is optimal when . Thus, we can obtain

The influence of the mutation probability on the optimization performance was investigated. The chromosome size and the maximum number of epochs are set at 10 and 100, respectively. Let , such that .

The magnitude of the rotational angle and the mutation probability is set as follows:

To determine the relationship between mutation probability and rotational angle as well as to avoid the influence of random factors, we performed the calculation 10 times by using different mutation probabilities in each rotational angle from to and obtained the average value as the optimization result. The optimization result is shown in Figures 10 and 11.

Figures 10 and 11 show that the mutation operator can accelerate the optimization process when . The optimal optimization effect can be obtained when . In general, the range of can be selected from 0.01 to 0.5.

By synthesizing the description, the optimum parameters for FCQIEA can be expressed as follows:(1);(2);(3);(4).

5.2. Comparison of the Different Chains of QIEA

In this section, we tested the performance of the proposed FCQIEA by using two application examples of the extremum optimizing complex functions (the Goldstein-Price and the Shubert functions). We also compared FCQIEA with CQIEA, DCQIEA, and TCQIEA in the experiments.

In the following section, we evaluated the optimization performance of FCQIEA and compared it with that of CQIEA, DCQIEA, and TCQIEA. In these three algorithms, the number of chromosomes, the mutation probability, and the maximum number of epochs are set at 20, 0.04, and 100, respectively. In FCQIEA, , , and are . In TCQIEA, and are [50]. In DCQIEA, is . In CQIEA, each variable of the Shubert function is described by 25 binary digits, where the first bit represents the sign, the second to fifth bits represent the integral part of the variable, and the sixth to twenty-fifth bits represent the fraction part of the variable. In this case, each chromosome consists of 50 qubits. For the Goldstein-Price function, each variable is described by 18 binary digits, where the first bit represents the sign, the second and third bits represent the integral part of the variable, and the fourth to eighteenth bits represent the fraction part of the variable [50]. Therefore, each chromosome consists of 36 qubits. Two functions are optimized 10 times by FCQIEA, TCQIEA, DCQIEA, and CQIEA. The optimization results are shown in Figures 12 and 13 as well as in Table 2.


AlgorithmGoldstein-Price functionShubert function
Optimal resultWorst resultAverage resultConvergence timesAverage times (s)Optimal resultWorst resultAverage resultConvergence resultAverage times (s)

CQIEA 3.1312 3.28523.214205.02059.98289.89379.927004.8763
DCQIEA3.00473.13603.041841.31269.99809.93789.9739 10.7487
TCQIEA3.00703.01523.0070102.26939.99979.96439.9916 61.0335
FCQIEA 3.00003.00593.001610 2.24042101010101.1191

Table 2 and Figures 12 and 13 show that FCQIEA is the most efficient algorithm and the optimized result is considered optimal. TCQIEA and DCQIEA are also considered efficient algorithms. By contrast, our results revealed that CQIEA is the most inefficient of the three algorithms.

Table 2 and Figures 12 and 13 also show that FCQIEA is the most efficient and the optimized result is optimal. TCQIEA and DCQIEA are also considered efficient algorithms. By contrast, our results revealed that CQIEA is the most inefficient of the three algorithms. In FCQIEA, four-chain encoding method enhances the search capability compared with TCQIEA and DCQIEA. The application of mutation operator avoids running into the local minimum. The TCQIEA efficiency is lower than FCQIEA because FCQIEA have four gene chains, but TCQIEA and DCQIEA have less than three gene chains. For the same reason, the efficiency of DCQIEA is lower than that of TCQIEA. The running time of CQIEA is the greatest because of frequently encoding and decoding the binary number. Given that the optimization solution is obtained by probability, CQIEA is not the best choice to solve continuous optimization problems. Thus, we can improve the QEA efficiency by adding gene chains.

5.3. Comparison of Other QIEAs

Five benchmark numerical optimization problems, including Ackley (F1), Griewank (F2), Rastrigin (F3), Schwefel (F4), and Rosenbrock (F5), are described in Table 3.


FunctionFunction expressionSolution space

Ackley 0
Griewank 0
Rastrigin 0
Schwefel 0
Rosenbrock 0

The multimodal functions include Ackley, Griewank, Rastrigin, and Schwefel. The Ackley function has a surface with numerous local optima because of its exponential term. The variables of Griewank function exhibit interdependence because the function has a product term. Similarly, Rastrigin function has numerous local minima. The surface of Schwefel function is composed of numerous peaks and valleys. The best minimum among the functions is located far from the global minimum, which is found near the boundaries of the search domain [48, 76]. Rosenbrock is a continuous, convex, and unimodal function. Given that the global optimum is inside a long, narrow, parabolic-shaped flat valley and the variables are dependent, the gradients generally do not point toward the optimum and these gradients are difficult to converge at the global optimum. The functions are used to compare the different QIEA performances, including bQIEA and rQIEA, by using 30 dimensions. The population size is set at 50 for FCQIEA. The stopping criterion is the number of function evaluations. Each test function is performed with 50 independent runs. The mean best values and the standard deviations are recorded for each test function. Table 3 lists the simulation test results, in which FCQIEA provides better results than the two other versions of QIEA, which are used to search for optimal solutions and maintain robustness. The number of runs is 50. Mean, Std., and NoFE represent the best mean, the standard deviation, and the number of function evaluations, respectively. The bold style highlights the best result for each function (Table 4).


ProblemsF1F2F3 F4F5

NoFE
bQIEAMean
Std.
rQIEAMean
Std.
FCQIEAMean