Using Augmented Lagrangian Particle Swarm Optimization for Constrained Problems in Engineering

Abstract

The general nonlinear optimization problem is given by the nonlinear objective function f, which is to be minimized with respect to the design variables x and the nonlinear equality and inequality constraints. This can be formulated by

$$ \mathop {minimize}\limits_p \psi (p), p \in \mathbb{D} \cap \mathbb{F}, \mathbb{D} \subseteq \mathbb{R}^n $$

(12.1)

subject to the nonlinear equality and inequality constraints

$$ g(p) = 0, g :\mathbb{R}^n \to \mathbb{R}^{m_e } , $$

(12.2)

$$ h(p) \leqslant 0, h :\mathbb{R}^n \to \mathbb{R}^{m_i } $$

(12.3)

which define the feasible region $\mathbb{F}$ and the search space $\mathbb{D}$ is additionally bounded by the simple bounds

$$ p_l \leqslant p \leqslant p_u $$

(12.4)

The following chapter contains in a modified and unified way material which was published before by the authors in Structural and Multidisciplinary Optimization (Sedlaczek, Eberhard, 2006).