Using augmented Lagrangian particle swarm optimization for constrained problems in engineering

The comparatively new stochastic method of particle swarm optimization (PSO) has been applied to engineering problems especially of nonlinear, non-differentiable, or non-convex type. Its robustness and its simple applicability without the need for cumbersome derivative calculations make PSO an attractive optimization method. However, engineering optimization tasks often consist of problem immanent equality and inequality constraints which are usually included by inadequate penalty functions when using stochastic algorithms. The simple structure of basic particle swarm optimization characterized by only a few lines of computer code allows an efficient implementation of a more sophisticated treatment of such constraints. In this paper, we present an approach which utilizes the simple structure of the basic PSO technique and combines it with an extended non-stationary penalty function approach, called augmented Lagrange multiplier method, for constraint handling where ill conditioning is a far less harmful problem and the correct solution can be obtained even for finite penalty factors. We describe the basic PSO algorithm and the resulting method for constrained problems as well as the results from benchmark tests. An example of a stiffness optimization of an industrial hexapod robot with parallel kinematics concludes this paper and shows the applicability of the proposed augmented Lagrange particle swarm optimization to engineering problems.