Backtracking biogeography-based optimization for numerical optimization and mechanical design problems

Abstract

As a novel Evolutionary Algorithm (EA), Biogeography-Based Optimization (BBO), inspired by the science of biogeography, draws much attention due to its significant performance in both numerical simulations and practical applications. In BBO, the features in poor solutions have a large probability to be replaced by the features in good solutions. The replacement operator is termed migration. However, the replacement causes a loss of the features in poor solutions, breaks the diversity of population and may lead to a local optimal solution. To overcome this, we design a novel migration operator to propose Backtracking BBO (BBBO). In BBBO, besides the regular population, an external population is employed to record historical individuals. The size of external population is the same as the size of regular population. The external population and regular population are used together to generate the next population. After that, the individuals in external population are randomly selected to be updated by the individuals in current population. In this way, the external population in BBBO can be considered as a memory to take part in the evolutionary process. The memory takes into account both current and historical data to generate next population, which enhances algorithm’s ability in exploring searching space. In numerical simulation, 14 classical benchmarks are employed to test BBBO’s performance and several classical nature inspired algorithms are use in comparison. The results show that the strategy in BBBO is feasible and very effective to enhance algorithm’s performance. In addition, we apply BBBO to mechanical design problems which involve constraints in optimization. The comparison results also exhibit that BBBO is very competitive in solving practical optimization problems.

Appendix:: Benchmark functions

The benchmarks used in Section 4 are given as follows. In the expressions, D is the dimension of the decision variable. In Section 4, the value of D is set as 20.

• A. Ackley’s Function:

  $$\begin{array}{@{}rcl@{}} f_{1}\! =\! &-&20exp\left( -0.2\sqrt{\frac{1}{D}{\sum}_{i=1}^{D}{x_{i}^{2}}}\right) -exp\left( \frac{1}{D}{\sum}_{i=1}^{D}\cos2\pi x_{i}\right)\\ &+& 20 + e,~~-32\leq x_{i}\leq 32, \end{array} $$

• B. Fletcher-Powell:

  $$\begin{array}{ll} &f_{2}(x) \,=\, {\sum}_{i=1}^{D}\left( A_{i}-B_{i}\right)^{2}.\\ \text{where}\; A_{i}={\sum}_{j=1}^{D}(a_{ij}\sin\alpha_{j} + b_{ij}\cos\alpha_{j}),&\\ B_{i}={\sum}_{j=1}^{D}(a_{ij}\sin x_{j} + b_{ij}\cos x_{j}),&\\ x_{i},\alpha_{i} \in [-\pi,\pi],&\\ a_{ij}, b_{ij} \in [-100,100].& \end{array} $$

• C. Generalized Griewanks Function:

  $$\begin{array}{@{}rcl@{}} f_{3}(x)=\frac{1}{4000}{\sum}_{i=1}^{D}{x_{i}^{2}}-{\prod}_{i=1}^{D}\cos\left( \frac{x_{i}}{\sqrt{i}}\right)+1,~~\\-600\leq x_{i} \leq 600, \end{array} $$

• D. Penalized Function 1:

  $$\begin{array}{@{}rcl@{}} f_{4}(x) \!&=&\! \frac{\pi}{D}\!\left\{\!10\sin^{2}(\pi y_{1})\,+\,{\sum}_{i=1}^{D-1}\!(y_{i}-1)^{2}\!\left( 1+10\sin^{2}(\pi y_{i+1})\right)\right\}\\ &&+{\sum}_{i=1}^{D} u(x_{i},10,100,4),~~-50\leq x_{i} \leq 50, \end{array} $$
  $$\begin{array}{@{}rcl@{}} \text{where}\; u(x_{i},a,k,m) &=& \left\{\begin{array}{ll} k(x_{i}-a)^{m} & x_{i}>a,\\ 0 & -a \leq x_{i} \leq a \\ k(-x_{i}-a)^{m} &x_{i}<-a \end{array}\right.\\ \text{and}\, y_{i} &=& 1 + \frac{1}{4}(x_{i}+1), \end{array} $$

• E. Penalized Function 2:

  $$\begin{array}{@{}rcl@{}} f_{5}(x)\,=\,0.1\left\{\sin^{2}(\pi 3x_{1})+{\sum}_{i=1}^{D-1}\!(x_{i}-1)^{2} \left[1+\sin^{2}(x\pi x_{i+1})\right]\right.\\\left.{\phantom{\frac{2}{5}}}+\!(x_{n}-1)^{2}[1+\sin^{2}(2\pi x_{D})]\right\}\,+\,{\sum}_{i=1}^{D}u(x_{i},5,100,4), \end{array} $$
  $$\begin{array}{@{}rcl@{}} \text{where}\; -50\leq x_{i} \leq 50, \;y_{i} = 1 + \frac{1}{4}(x_{i}+1), \\ \text{and}\; u(x_{i},a,k,m) = \left\{\begin{array}{ll} k(x_{i}-a)^{m} & x_{i}>a,\\ 0 & -a \leq x_{i} \leq a \\ k(-x_{i}-a)^{m} &x_{i}<-a \end{array}\right. \end{array} $$

• F. Quartic Function:

  $$\begin{array}{@{}rcl@{}} f_{6}(x) &=& {\sum}_{i=1}^{D}i{x_{i}^{4}} + random [0,1),\\ &&-1.28\leq x_{i} \leq1.28, \end{array} $$

• G. Generalized Rastrigins Function:

  $$\begin{array}{@{}rcl@{}} f_{7}(x) &=& {\sum}_{i=1}^{D}\left[x_{i}^{2}-10\cos(2\pi x_{i})+10\right],\\ &&-5.12\leq x_{i} \leq 5.12, \end{array} $$

• H. Generalized Rosenbrocks Function:

  $$\begin{array}{@{}rcl@{}} f_{8}(x)&=&{\sum}_{i=1}^{D-1}\left[100(x_{i+1}-x_{i}^{2})^{2} + (x_{i}-1)^{2}\right],\\ &&-30\leq x_{i}\leq 30, \end{array} $$

• I. Schwefel’s Problem 1.2:

  $$f_{9}(x) = {\sum}_{i=1}^{D}\left( {\sum}_{j=1}^{i} x_{j}\right)^{2},~~-100\leq x_{i}\leq 100, $$

• J. Schwefel’s Problem 2.21:

  $$f_{10}(x)= \max\limits_{i}\left\{|x_{i}|,1\leq i \leq D\right\},~~-100\leq x_{i}\leq100, $$

• K. Schwefel’s Problem 2.22:

  $$f_{11}(x)= {\sum}_{i=1}^{D}|x_{i}|+{\prod}_{i=1}^{D}|x_{i}|,~~~-10\leq x_{i}\leq 10, $$

• L. Schwefel’s Problem 2.26:

  $$f_{12}(x) = -{\sum}_{i=1}^{D}\left( x_{i}\sin\left( \sqrt{|x_{i}|}\right)\right),~~-500\leq x_{i}\leq500, $$

• M. Sphere Model:

  $$f_{13}(x)={\sum}_{i=1}^{D}{x_{i}^{2}},~~100\leq x_{i}\leq 100, $$

• N. Step Function:

  $$f_{14}(x)={\sum}_{i=1}^{D}(\lfloor x_{i}+0.5\rfloor)^{2},~~-100\leq x_{i}\leq100, $$