Population recombination strategies for multi-objective particle swarm optimization

Abstract

Multi-objective particle swarm optimization algorithm (MOPSOs) has been found to exhibit fast convergence speed but with high probability to fall into local optimum. To overcome this shortcoming, a population recombination strategy is combined with a new mutation strategy to strengthen the ability to jump out of local optimum. From the investigation conducted, it can be found that, when the MOPSO falls into local optimum, the population will stop producing effective particles to update the archive. Population recombination strategy, which utilizes the information of the best variable found so far to construct the new population. This can increase the probability for population to approach the Pareto optimal front, while additional mutation operation can enhance the diversity of population. Experimental study on the bi-objective and three-objective benchmark problems shows that the MOPSO based on proposed strategies is superior to previous multi-objective algorithms in the literature.

Appendix 1: Performance measures

The three performance indicators used in this paper are described as follows.

Generational distance (GD): The GD metric is defined as:

$$\begin{aligned} \textit{GD}(A,P)=\frac{\sum \nolimits _{i=1}^{\vert A\vert } {d(A_i ,P)} }{\vert A\vert } \end{aligned}$$

where A is the archive found by the algorithm and P is the reference point set (500 point in the experiment) in PF generated prior. $d_{i}$ is the minimum distance between the $A_{i}$ and the point in P. Obviously, this value can only measure the distance from the archive to the PF, and the distribution of the archive cannot be evaluated. Therefore, the $\Delta$ indicator is employed for the comprehensive evaluation of the algorithm together with GD.

Diversity indicator: $\Delta$ measures the extreme and spread among the obtained solutions:

$$\begin{aligned} \Delta =\frac{d_f +d_l +\sum \nolimits _{i=1}^{N-1} {\vert d_i -{\bar{d}} \vert } }{d_f +d_l +(N-1) {\bar{d}} } \end{aligned}$$

Here, $d_{i}$ is Euclidean distance between nearby solutions in A and $d^{-}$ is the average of these distances. $d_{f}$ and $d_{l}$ are the Euclidean distances between the extreme solutions and the boundary solutions of the A. N is the number of solutions in A.

Inverted generational distance (IGD): The IGD metric is defined as:

$$\begin{aligned} \textit{IGD}(P,A)=\frac{\sum \nolimits _{i=1}^{\vert P\vert } {d(P_i ,A)} }{\vert P\vert } \end{aligned}$$

Here the definition of P and A is the same as those for GD indicator, but $d_{i }$is the closest distance from the point in P to the point in A. It can be used to evaluate both convergence and diversity performance simultaneously.