Firefly penalty-based algorithm for bound constrained mixed-integer nonlinear programming

In this article, we aim to extend the firefly algorithm (FA) to solve bound constrained mixed-integer nonlinear programming (MINLP) problems. An exact penalty continuous formulation of the MINLP problem is used. The continuous penalty problem comes out by relaxing the integrality constraints and by adding a penalty term to the objective function that aims to penalize integrality constraint violation. Two penalty terms are proposed, one is based on the hyperbolic tangent function and the other on the inverse hyperbolic sine function. We prove that both penalties can be used to define the continuous penalty problem, in the sense that it is equivalent to the MINLP problem. The solutions of the penalty problem are obtained using a variant of the metaheuristic FA for global optimization. Numerical experiments are given on a set of benchmark problems aiming to analyze the quality of the obtained solutions and the convergence speed. We show that the firefly penalty-based algorithm compares favourably with the penalty algorithm when the deterministic DIRECT or the simulated annealing solvers are invoked, in terms of convergence speed.