Improvement of the performance of the Quantum-inspired Evolutionary Algorithms: structures, population, operators

Abstract

Population diversity is very important in giving the algorithm the power to explore the search space and not get trapped in local optima. In this respect, using a probabilistic representation for the quantum individuals, the Quantum-inspired Evolutionary Algorithms (QiEA) claim higher diversity in the population. Here, considering this important feature of QiEA, we propose different structures to offer better interaction between the q-individuals and propose new operators to preserve the diversity in the population and thus improve the performance of the QiEA. The effect of the structured population is investigated on the performance of the algorithm. Additionally, two operators are proposed in this paper. Being called the Diversity Preserving QiEA the first operator finds the converged similar q-individuals around a local optimum and while keeping the best q-individuals, by reinitializing the inferior ones pushes them out of the basin of attraction of the local optimum, so helping the algorithm to search other regions in the search space. The other operator is a reinitialization operator which by reinitializing the whole population helps it escape from the local optima it is trapped in. By studying the effect of the parameters of the proposed operators on their performance we show how the proposed operators improve the performance of QiEA. Experiments are performed on Knapsack, Trap and fourteen numerical objective functions and the results show better performance for the proposed algorithm than the original version of QiEA.

Appendix

In this section we describe the benchmark problems we have used in this paper.

1.1 Trap problem

The Trap problem is defined as,

$$\begin{aligned} f(x)=\sum _{i=0}^{N-1} {\rm{Trap}}(x_{5i+1},x_{5i+2},x_{5i+3},x_{5i+4},x_{5i+5}), \end{aligned}$$

(18)

where $N$ is the number of traps and equals the size of the problem divided by 5. A Trap is defined as,

$$\begin{aligned} {\rm{Trap}}={\left\{ \begin{array}{ll}4-{\rm{ones}}(x),& {\rm{if}}\;\;{\rm{ones}}(x)\le 4\\ 5,& {\rm{if}}\;\;{\rm{ones}}(x)=5\end{array}\right. } \end{aligned}$$

(19)

where the function “ones” returns the number of ones in the binary string $x$. The Trap problem has a local optimum at $(0,0,0,0,0)$ and a global optimum at $(1,1,1,1,1)$.

1.2 Knapsack problem

The Knapsack problem is a well-known combinatorial optimization problem which belongs to the class of NP-Hard problems. The Knapsack problem can be described as selecting a set of items $x_i$, $i=1,2,\ldots ,m$ with profits $p_i$, and weights $w_i$ for a knapsack with capacity $C$. Given a set of $m$ items and a knapsack with capacity $C$, select the subset of the items to maximize the profit function $f(x)$,

$$\begin{aligned} f(x)=\sum _{i=1}^m{p_i x_i}, \end{aligned}$$

(20)

such that,

$$\begin{aligned} \sum _{i=1}^m{w_i x_i}\le C. \end{aligned}$$

(21)

In this paper we consider

$$\begin{aligned} w_i={\rm R} (1,v),\;\;\;p_i= {\rm R} (1,v), \end{aligned}$$

(22)

where ${\rm R (.,.)}$ is a uniform random number generator, $v=10$ and the knapsack capacity is determined as,

$$\begin{aligned} C=\frac{1}{2}\sum _{i=1}^m w_i. \end{aligned}$$

(23)

Two types of QiEA methods are used in this paper to test our proposed algorithms [13]: the penalty function and the repair method.

In the QiEA based on penalty functions, a binary of the length $m$ represents a chromosome $x$ which is a solution to the problem. The profit $f(x)$ of each string is determined as

$$\begin{aligned} f(x)=\sum _{i=1}^m p_i x_i -{\rm{Pen}}(x), \end{aligned}$$

(24)

where Pen$(x)$ is a penalty function. There are different ways of assigning the penalty function [13]. Here we use two types of penalties: the logarithmic and the liner penalty,

$$\begin{aligned} {\rm{Pen}}_1 (x)&= \log _2 \left( 1+\rho \left( \sum _{i=1}^m w_i x_i -C\right) \right) ,\end{aligned}$$

(25)

$$\begin{aligned} {\rm{Pen}}_2 (x)&= \rho \left( \sum _{i=1}^m w_i x_i -C\right) , \end{aligned}$$

(26)

where $\rho$ is $\max _{i=1}^m\{p_i/w_i\}$.

In QiEA based on repair methods, the profit $f(x)$ of each string is determined as,

$$\begin{aligned} f(x)=\sum _{i=1}^m{p_i x'_i}, \end{aligned}$$

(27)

where $x'$ is a repaired vector of the original vector $x$. The only difference between the two types of the repairs is the way the items are removed from the knapsack:

$\hbox {Rep}_1$ :

(random repair): In this method the elements are selected randomly and then removed from the knapsack.

$\hbox {Rep}_2$ :

(greedy repair): The items in the knapsack are sorted in a decreasing order based on their profit to weight ratio. The selection procedure then chooses the last item for deletion.

1.3 Numerical functions

Global numeric optimization problems arise in many fields of science, engineering, and business [48]. Here we use 14 benchmark maximization functions for testing our proposed algorithm. The $f_1{-}f_8$ are multimodal functions where the number of local optima increases with the problem dimension. For example $f_1$ has about $6^n$ local optima in $[-500,500]^n$ and $f_2$ has about $10^n$ local optima in the search space of $[-5.12,5.12]^n$.

Generalized Schwefel’s Function 2.26:

$$\begin{aligned} f_1(x)=\sum ^n_{i=1}\left( -x_i \sin \sqrt{\left| x_i\right| }\right),\quad U=[-500,500]^n. \end{aligned}$$

Generalized Rastrigin’s Function:

$$\begin{aligned} f_2(x)=\sum ^n_{i=1}x^2_i-10 \cos \left( 2\pi x_i\right) +10,\quad U=[-5.12,5.12]^n. \end{aligned}$$

Ackley’s Function:

$$\begin{aligned} f_3(x)=-20 \exp \left( -0.2\sqrt{\frac{1}{n}\sum ^n_{i=1}x^2_i}\right) - \exp \left( \frac{1}{n}\sum ^n_{i=1}\cos (2\pi x_i)\right) \\+20+e,\quad U=[-32,32]^n. \end{aligned}$$

Generalized Griewank Function:

$$\begin{aligned} f_4(x)=\frac{1}{4{,}000}\sum ^n_{i=1}x^2_i-\prod ^n_{i=1}\cos \left( \frac{x_i}{\sqrt{i}}\right) +1,\quad U=[-600,600]^n. \end{aligned}$$

Generalized Penalized Function 1:

$$\begin{aligned} f_5(x)=&\frac{\pi }{n}\{10\sin ^2(\pi y_1)+\sum ^{n-1}_{i=1}(y_i-1)^2 \times [1+10\sin ^2(\pi y_{i+1})]\\&+(y_n-1)^2\}+\sum ^n_{i=1}u(x_i,10,100,1)\\&{\left\{ \begin{array}{ll} k(x_i-a)^p & x_i>a \\ 0 & -a\le x_i\le a\\ k(-x_i-a)^p & x_i<-a\end{array}\right. }\\ y_i=&1+\frac{1}{4}(x_i+1),\quad U=[-50,50]^n. \end{aligned}$$

Generalized Penalized Function 2:

$$\begin{aligned} f_6(x)=&\frac{1}{10}\{\sin ^2(3\pi x_1)+\sum ^{n-1}_{i=1}(x_i-1)^2\left[ 1+\sin ^2(3\pi x_{i+1})\right] \\&+(x_n-1)^2\left[ 1+\sin ^2(2 \pi x_n)\right] \}+\sum ^n_{i=1}u(x_i,5,100,1),\quad U=[-50,50]^n. \end{aligned}$$

Michalewicz Function:

$$\begin{aligned} f_7(x)=-\sum ^n_{i=1}\left( \sin (x_i)\times \left( \sin \left( \frac{ix^2_i}{\pi }\right) \right) ^{2n}\right),\quad U=[-\pi ,\pi ]. \end{aligned}$$

Goldberg & Richardson Function:

$$\begin{aligned} f_8(x)=&\sum _{i=1}^n g_i h_i,\quad g_i=[\sin (5\pi x_i+0.5)]^2,\\ h_i=&\exp \left( -2.0\log (2.0)\frac{(x_i-0.1)^2}{0.64}\right) ,\quad S=[-1,1]^n. \end{aligned}$$

Sphere Model:

$$\begin{aligned} f_9(x)=\sum _{i=1}^n x^2_i,\quad U[-100,100]^n. \end{aligned}$$

Schwefel’s Function 2.22:

$$\begin{aligned} f_{10}(x)=\sum ^n_{i=1}\left| x_i\right| +\prod ^n_{i=1}\left| x_i\right| ,\quad U=[-10,10]^n. \end{aligned}$$

Schwefel’s Function 2.21:

$$\begin{aligned} f_{11}(x)=\max _i\left\{ {|x_i|,1\le i\le n}\right\} ,\quad U=[-100,100]^n. \end{aligned}$$

Dejong Function 4:

$$\begin{aligned} f_{12}(x)=\sum _{i=1}^{n}ix^4_i+{\rm{Gauss}}(0,1), \quad U=[-10,10]^n. \end{aligned}$$

Rosenbrock Function:

$$\begin{aligned} f_{13}(x)=\sum _{i=1}^{n-1}100\left( x_{i+1}-x_i^2\right) ^2+\left( 1-x_i\right) ^2,\quad U=[-2,2]^n. \end{aligned}$$

Kennedy multimodal function generator:

$$\begin{aligned} f_{14}(x)=&\min _{j=1}^n\left( \sum _{i=1}^n\left[ \frac{1}{1+\exp (-x_i)}\right] ^2+\frac{(j-1)^{0.15}}{15}\right) ,\\ U=&[-4,4]^n. \end{aligned}$$

Where $U$ is the search space and n is the problem dimension. These functions have several local minima and one global minimum. Since they are used for maximization process, we use it with a negative sign change $f(x)$ for fitness function.