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.

中文翻译：

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使用PSO-SVM进行结构优化的非惩罚性机器学习约束处理

首先提出解决无约束优化问题的方法，采用一些惩罚函数代表了用超启发式粒子群优化算法（PSO）解决约束问题的常用方法。本文采用监督分类机学习方法，即支持向量机（SVM），实现了一种新的基于非惩罚度的PSO约束处理方法。由于其通用性，使用SVM进行的约束处理似乎对非线性和不连续边界都更加适应。为了保留种群的可行性，还实施了简单的二等分算法。为了提高算法的搜索性能，还采用了约束的松弛函数。在本文的最后，讨论了两个数值文献基准实例和两个结构实例。第一个结构示例是指均匀截面恒定的简支梁。第二个是优化平面简单支撑的沃伦桁架梁的优化。就目标函数而言，所获得的结果表明，即使在结构优化领域中，该新方法仍是解决约束优化问题的有效替代方法。此外，正如沃伦桁架梁问题所证明的那样，该新算法提供了一种最佳的结构设计，从技术角度来看，这也是一个很好的解决方案，并且舍入不明显地影响了优化设计过程。第二个是优化平面简单支撑的沃伦桁架梁的优化。就目标函数而言，所获得的结果表明，即使在结构优化领域中，该新方法仍是解决约束优化问题的有效替代方法。此外，正如沃伦桁架梁问题所证明的那样，该新算法提供了一种最佳的结构设计，从技术角度来看，这也是一个很好的解决方案，并且舍入后的舍入不会显着损害优化设计过程。第二个是优化平面简单支撑的沃伦桁架梁的优化。就目标函数而言，所获得的结果表明，即使在结构优化领域中，该新方法仍是解决约束优化问题的有效替代方法。此外，正如沃伦桁架梁问题所证明的那样，该新算法提供了一种最佳的结构设计，从技术角度来看，这也是一个很好的解决方案，并且舍入不明显地影响了优化设计过程。