A two-stage quantum-behaved particle swarm optimization with skipping search rule and weight to solve continuous optimization problem

Abstract

Quantum-behaved particle swarm optimization (QPSO) is a recently developed heuristic method by particle swarm optimization (PSO) algorithm based on quantum mechanics, which outperforms the search ability of original PSO. But as many other PSOs, it is easy to fall into the local optima for the complex optimization problems. Therefore, we propose a two-stage quantum-behaved particle swarm optimization with a skipping search rule and a mean attractor with weight. The first stage uses quantum mechanism, and the second stage uses the particle swarm evolution method. It is shown that the improved QPSO has better performance, because of discarding the worst particles and enhancing the diversity of the population. The proposed algorithm (called ‘TSQPSO’) is tested on several benchmark functions and some real-world optimization problems and then compared with the PSO, SFLA, RQPSO and WQPSO and many other heuristic algorithms. The experiment results show that our algorithm has better performance than others.

Appendix

1. F1 Sphere’s function

$$f_{1} = \sum\limits_{i = 1}^{n} {x^{2}_{i} }$$

2. F2 Schwefel’s problem 2.22

$$f_{2} (x) = \sum\limits_{i = 1}^{n} {|x^{2}_{i} |} + \prod\limits_{i = 1}^{30} {|x_{i} } |$$

3. F3 Schwefel’s problem 1.2

$$f_{3} (x) = \sum\limits_{i = 1}^{N} {\left( {\sum\limits_{j = 1}^{i} {x_{j} } } \right)}^{2}$$

4. F4 Schwefel’s problem 2.21

$$f4(x) = \hbox{max} \{ \left| {x_{i} } \right|\} ,i = 1,2,{ \ldots },n$$

5. F5 Generalized Rosenbrock’s function

$$f5(x) = \sum\limits_{i = 1}^{n - 1} {100 \cdot (x_{i + 1} - x_{i}^{2} )^{2} + (1 - x_{i} )^{2} }^{{}}$$

6. F6 Step function

$$f6(x) = \sum\limits_{i = 1}^{n} {\left( {\left\lfloor {x_{i} } \right\rfloor + 0.5} \right)^{2} }$$

7. F7 Quartic function, i.e., noise

$$f7(x) = \sum\limits_{i = 1}^{n} {ix^{4}_{i} } + rnd\left[ {0,1} \right)$$

8. F8 Generalized Griewank’s function

$$f8(x) = \frac{1}{4000}\sum\limits_{i = 1}^{n} {x_{i}^{2} } - \prod\limits_{i = 1}^{n} {\cos (\frac{{x_{i} }}{\sqrt i })} + 1$$

9. F9 Generalized Rastrigin’s function

$$f9(x) = \sum\limits_{i = 1}^{n} {(x_{i}^{2} - 10\cos (2\pi x_{i} ) + 10)}$$

10. F10 Ackley’s function

$$f10(x) = - 20\exp \left( { - 0.2\sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {x_{i}^{2} } } } \right) - \exp \left( {\frac{1}{n}\sum\limits_{i = 1}^{n} {\cos (2\pi x_{i} )} } \right) + 20 + e$$

11. F11 Shaffer’s function

$$f11(x) = 0.5 + \frac{{(\sin ({\text{sqrt}}(x_{1} )^{2} + x_{2}^{2} ))^{2} - 0.5}}{{1 + 0.00.(x_{1}^{2} + x_{2}^{2} )^{2} }}$$

12. F12 Branin’s function

$$f12(x) = \left(x_{2} - \frac{5.1}{{4\pi^{2} }}x_{1}^{2} + \frac{5}{\pi }x_{1} - 6 \right)^{2} + 10\left(1 - \frac{1}{8\pi}\right)\cos x_{1} + 10$$