In this chapter, we have introduced the basic definitions, notations, and concepts as- sociated with multi-objective optimization, particle swarm optimization and quantum
behaved particle swarm optimization. Then we presented an overview of the different multi-objective evolutionary algorithms that have been developed and successfully ap- plied to solve multi-objective optimization problems. We emphasized the PSO-Pareto based methods because these are most relevant for the remainder of the thesis, and they represent the current state of art. The chapter ends with a summary of the important strengths and weaknesses of the different MOP approaches.
In the following chapter, a new addition to the canyon of swarm based multi-objective optimization methods will be devised. Following the same spirit of MOPSO, we propo- se an extension of QPSO to solve continuous multi-objective optimization problems that aims to achieve better convergence and diversity simultaneously. Our new approach is ter- med in the following as Multi Objective Quantum-behaved Particle Swarm Optimization
KAPITEL
3
A New Framework for MOP: Multi-Objective
Quantum-behaved Particle Swarm Optimization
(MOQPSO) for Unconstrained Problems
In this chapter,1 we propose a new approach for multi-objective optimization based on
QPSO. In particular, we show how we extended QPSO, and developed it to handle un- constrained multi-objective optimization problems. This extension includes a definition of a leader selection strategy, a policy to maintain the Pareto set during the search process and an overall dynamics that helps evolving an initial Pareto set towards the optimal one. Specifically, we address the way the global best solutions are recorded within an archive and used to compute the local attractor point of each particle.
1A shorter version of the work in this chapter has been published in the following:
Heyam Al-Baity, Souham Meshoul, and Ata Kaban. On Extending Quantum Behaved Particle Swarm Optimization to Multi-objective Context. In Proceedings of the IEEE World Congress on Computational Intelligence (IEEE CEC 2012), pp. 1-8, 2012.
3.1
Main Features of the Proposed MOQPSO
The typical dynamics of a multi-objective Pareto based swarm algorithm include two main phases. In the first phase, the initial swarm is generated, the algorithm parameters are set and an initial set of non-dominated solutions is derived. The second phase is generally an iterative procedure during which the positions of particles are recomputed according
to equations (2.3) and (2.4) as explained in section 2.2.2. Self best performance of each
particle is updated along with the current Pareto set of non-dominated solutions. There- fore, extending QPSO to multi-objective optimization should be done in a way to find a Pareto front as close as possible to the optimal one while ensuring a uniform distri- bution of the non-dominated solutions within it. We refer to the proposed approach as MOQPSO (Multi-Objective Quantum-behaved Particle Swarm Optimization). Generally, in single objective QPSO, only one self best particle and one global best particle have to be considered when updating particles’ positions. In a multi-objective context, a set
of non-dominated solutions should be handled. Recall the main equation (2.3) in section
2.2.2, that governs the move of particles in the search space:
xt+1ij = ptij ± β.|mbesttj − xtij|. ln(1/utij)
We can identify from this equation, in a direct way, the two main channels through
which the extension of QPSO to MOP should be studied, namely: the local attractor (pt
ij) and the mean best (mbest). In an indirect way, the impact of the contraction expansion parameter β has to be investigated as well. Therefore, the computation of each particle position requires three main components namely:
• The local attractor point of a particle. As given in equation (2.4) in section 2.2.2:
pt
ij = ϕtij.sbesttij + (1 − ϕtij).gbesttj with ϕtij = rand(0, 1)
It is calculated in a way that balances the influence of the self best performance of the particle and the global best performance of the entire swarm.
• The mean best position (also called mainstream thought by Sun et al. [105]), which
is obtained by averaging all the self best positions as shown in equation (2.5) in
section 2.2.2: mbestt= 1 N N P i=1sbesti = ( 1 N N P i=1sbesti1, 1 N N P i=1sbesti2, . . . , 1 N N P
i=1sbestiD) • The contraction expansion parameter β
Figure3.1below illustrates the key components that influence the design of multiobjective
QPSO (MQPSO).
Figur 3.1: The key components in MOQPSO design
Another way to highlight the main features of the proposed MOQPSO is through the differences between single objective QPSO and the proposed multi-objective QPSO
algorithm in terms of the number of self best and global best particles, the update strategy,
the selection strategy, and the mbest. These differences are summarized in Table 3.1.
Therefore, designing MOQPSO leads naturally to the following important research questions:
• How should the local attractor be updated for a given particle? • How should the mean best performance be derived?
• What is the impact of the contraction-expansion parameter (β) on the convergence of the algorithm?
• Which leader to select as the global best position for a particular particle to properly guide the particle’s navigation in a D-dimensional search space?
• Which archiving strategy to use to maintain the set of non-dominated solutions during the search process?