On a Hyperbolic Augmented Lagrangian Artificial Fish Swarm Based Method: Convergence Issues

where f : Rn → R and g : Rn → Rp are nonlinear continuous functions and Ω = {x ∈ Rn : −∞ < l ≤ x ≤ u < ∞}. Problems with equality constraints, h(x) = 0, can be reformulated into the above form by converting into a couple of inequality constraints h(x)− β ≤ 0 and −h(x)− β ≤ 0, where β is a small positive relaxation parameter. Since we do not assume that the objective function f is convex, the problem may have multiple optimal solutions in the feasible region. A global optimal solution is to be computed. For this class of optimization problems, methods based on penalty functions are quite common in the literature. In this type of methods, the constraint violation is combined with the objective function to define a penalty function, which aims at penalizing infeasible solutions by increasing their fitness values proportionally to their level of constraint violation. We are interested in a particular class of penalty functions known as augmented Lagrangian functions to handle the equality and inequality constraints of the problem (1). An augmented Lagrangian is a more sophisticated penalty function for which a finite penalty parameter value is sufficient to yield convergence to the solution of the constrained problem. We aim at analyzing the theoretical and practical behavior of an augmented Lagrangian algorithm that is constructed following the most common augmented Lagrangian paradigm [1, 2, 4]. The penalty function basis of our proposal is the exterior penalty function presented in [5], the so-called 2-parameter hyperbolic