An innovative hybrid multi-objective particle swarm optimization with or without constraints handling

Highlights

• A new hybrid optimizer is proposed in which an innovative local optimal particles search strategy, which on basis of particular analysis on disadvantage of global optimal particle method, is integrated into multi-objective particle swarm optimization.

• Select some non-dominated solutions lied in less-crowded region of external archive and make full use of optimal particles method to guide these solutions approach the Pareto front quickly.

• The multi-dimensional uniform mutation operator is performed to prevent algorithm from trapping into local optimum and a dynamic archive maintenance strategy is applied to improve the diversity of solutions.

• For coping with the constrained conditions consist in objective functions, we adopt an efficient infeasibility degree evaluation criterion to deal with these complex problems.

Abstract

This paper presents a new hybrid optimizer in which an innovative optimal particles local search strategy on basis of bound optimization by quadratic approximation (BOBYQA) algorithm and exterior penalty function method is integrated into particle swarm optimization (PSO). The main goal of the approach is to improve the convergence performance of PSO, and preserve the diversity of non-dominated set. Our algorithm selects some non-dominated solutions lied in less-crowded region of external archive based upon crowding distance value to construct a leader particles set, and make full use of optimal particles method to guide leader particles approach the Pareto front quickly. Meanwhile, a local optimal particles search strategy is proposed after particular analysis on disadvantage of global optimal particle search method, and names our algorithm as LOPMOPSO. Furthermore, the multi-dimensional uniform mutation operator is performed to prevent algorithm from trapping into local optimum, and a dynamic archive maintenance strategy is applied to improve the diversity of solutions. For coping with the constrained conditions consists in objective functions, we adopt an efficient infeasibility degree evaluation criterion to deal with these complex problems. Simulation results of various kinds of benchmark functions show that our approach is highly competitive in convergence speed and generates a well distributed and accurate set of non-dominated solutions easily. The solving of a 2-D aerodynamic optimization problem further validates its speed and effectiveness.

Introduction

Optimization problems can be classified as single objective optimization and multi-objective optimization according to the number of optimization objectives. Most real-world optimization problems that indwell in routine engineering application and scientific research can be regarded as multi-objective optimization (MO). Multi-objective optimization has a number of characteristics different from single objective optimization. The main difference is multi-objective optimization problems have a series of solutions instead of one. These solutions are called Pareto solutions set, which is regarded as equally good. Traditional multi-objective optimization methods always try to convert multiple objectives into one, such as linear weighting method and constraint method are the most common methods. Both methods obtaining a relatively satisfactory Pareto solution set easily. But linear weighting methods have the problem of allocating weighted coefficients and extremely rely on preferences of decision maker. Another disadvantage of this method is that it has difficulty to deal with the problem which possessing a non-convex Pareto front. The constraint method cannot handle those functions have complicated characteristics and it is frequently difficult to get satisfactory results, even no results show up.

More recently, an innovative global search method, particle swarm optimization (PSO) [1], based upon the interaction of individual entities called ‘particles’ proposed by Kennedy and Eberhart emerged in the 1995. The PSO is a heuristic algorithm inspired by the choreography of a bird flock and fish schooling. This algorithm has been found to be widely used in solving nonlinear continuous optimization tasks because of it has some attractive characteristics such as structure is simple, strong global search ability, few parameters, high convergence speed and easily to implement, etc. The special search mechanism of PSO makes it particularly suitable for multi-objective optimization. The first multi-objective optimization algorithm (MOPSO) [2] is reported by C. Coello in 2002. Since then, aims for obtaining a Pareto solutions set with the minimum deviation from true Pareto front and generating a well-distributed Pareto front [24], [25], researchers pay more and more attentions to study MOPSO and many excellent algorithms have been published in last decade [3], [4], [5], [6], [7], [8], [9], [10].

The researchers found that there are some drawbacks in MOPSO from practical applications. The most important of these problems is that MOPSO is easy to premature convergence [10] and poor local search ability. To overcome the disadvantage of MOPSO, the researchers incorporated some strategies into MOPSO to improve the diversity in Pareto optimal solutions and enhance the accuracy of solutions, such as introduce mutation to deal with premature convergence, the dynamic inertia weight to improve the local search ability of the PSO in the second half of the loop. These conventional methods have limited effect on improving the performance of MOPSO. In order to maintain balance between exploration and exploitation, some researchers introduce local search algorithm into MOPSO to cope with multi-objective problems and these strategies are highly effective to improve the performance of MOPSO. Liu [11] developed a hybrid PSO which combines the global search ability of particle swarm optimization with a synchronous local search heuristic for directed local fine-tuning. Santana [12] combined the high convergence rate of the particle swarm optimization algorithm with a local search approach based on scatter search, and proposed a new leader selection to accelerate convergence of PSO, then applied scatter search as a local search scheme, improving the spread of the non-dominated solutions found so far. Ono [13] proposed a hybridization of MOPSO and quasi-Newton method as an attempt to design effective memetic algorithm for robust optimization. Khoshahval [14] reported a new parallel optimization algorithm which combined PSO with simulated annealing method to deal with fuel management. Kaveh [15] proposed a new hybrid method for multi-objective optimization problem to improve the convergence and maintain diversity of solutions, charged system search method is incorporated into the search process of PSO to select the global best particle for every particle from Pareto set. Xu [16] proposed an efficient hybrid multi-objective particle swarm optimization with a multi-objective dichotomy line search. The algorithm used MOPSO to deal with premature convergence and diversity maintenance within the swarm, meanwhile, local search is periodically activated for fast local search to converge toward the Pareto front.

These hybrid algorithms effectively combined local search methods with MOPSO, and improve convergence performance of MOPSO partly. But experimental results also show that algorithms sometimes cannot solve multi-modality problem, sometimes easily leads to deterioration of the diversity of non-dominated solutions, some unable to deal with problems of the objective function gradient information cannot be directly calculated, or be incapable of improving the convergence precision significantly.

This paper introduces a new optimal particles strategy, called local optimal particles method, as local search approach, and an efficient hybrid MOPSO algorithm with local optimal particles method (LOPMOPSO) is proposed. First, this method uses the MOPSO to produce the non-dominated solutions then select some to construct a leader particles set, and establish a unique local optimal particle for each leader. BOBYQA algorithm [17] (for unconstrained problems) or exterior penalty function method (for constrained problems) is used to guide the non-dominated particles approach the local optimal particles. This method can protect the diversity of non-dominated solutions, and enhance the convergence speed and accuracy of the particle swarm optimization algorithm. Second, the mutation operator is applied to multi-dimensions of search space, which preventing particles from trapping into local optimum. Meantime, transforming all constraints into infeasibility degree(IFD), and evaluate infeasibility magnitude of particles in swarm with IFD, then employing IFD as the global and individual optimal particle selection criteria. Moreover, using crowding distance [21] mechanism to maintain the diversity of non-dominated solutions in the external archive.

Simulation results of benchmark functions show that our approach is highly competitive in convergence speed as well as obtaining a well distributed and accurate set of non-dominated solutions. LOPMOPSO also validated by a transonic airfoil aerodynamic optimization, we can achieve a better convergence to the Pareto front with adding our local search process. The remainder of the paper is organized as follows. In Section 2, some background information is described. Section 3, the principle of optimal particle local search and local particle construction are presented. In Section 4, the proposed algorithm is presented. Experimental results are discussed in Section 5. In Section 6, the application of a two dimensional aerodynamic optimization is given. Finally, Section 7 presents our conclusion and notes for future work.