Firstly formulated to solve unconstrained optimization problems, the common way to solve constrained ones with the metaheuristic particle swarm optimization algorithm (PSO) is represented by adopting some penalty functions. In this paper, a new nonpenalty-based constraint handling approach for PSO is implemented, adopting a supervised classification machine learning method, the support vector machine (SVM). Because of its generality, constraint handling with SVM appears more adaptive both to nonlinear and discontinuous boundary. To preserve the feasibility of the population, a simple bisection algorithm is also implemented. To improve the search performances of the algorithm, a relaxation function of the constraints is also adopted. In the end part of this paper, two numerical literature benchmark examples and two structural examples are discussed. The first structural example refers to a homogeneous constant cross-section simply supported beam. The second one refers to the optimization of a plane simply supported Warren truss beam. The obtained results in terms of objective function demonstrate that this new approach represents a valid alternative to solve constrained optimization problems even in structural optimization field. Furthermore, as demonstrated by the Warren truss beam problem, this new algorithm provides an optimal structural design which represents also a good solution from the technical point of view with a trivial rounding-off that does not jeopardize significantly the optimization design process.
A multi-objective constraint-handling method with PSO algorithm for constrained engineering optimization problems
