On a smoothed penalty-based algorithm for global optimization

This paper presents a coercive smoothed penalty framework for nonsmooth and nonconvex constrained global optimization problems. The properties of the smoothed penalty function are derived. Convergence to an $$\varepsilon $$ ε -global minimizer is proved. At each iteration k, the framework requires the $$\varepsilon ^{(k)}$$ ε ( k ) -global minimizer of a subproblem, where $$\varepsilon ^{(k)} \rightarrow \varepsilon $$ ε ( k ) → ε . We show that the subproblem may be solved by well-known stochastic metaheuristics, as well as by the artificial fish swarm (AFS) algorithm. In the limit, the AFS algorithm convergence to an $$\varepsilon ^{(k)}$$ ε ( k ) -global minimum of the real-valued smoothed penalty function is guaranteed with probability one, using the limiting behavior of Markov chains. In this context, we show that the transition probability of the Markov chain produced by the AFS algorithm, when generating a population where the best fitness is in the $$\varepsilon ^{(k)}$$ ε ( k ) -neighborhood of the global minimum, is one when this property holds in the current population, and is strictly bounded from zero when the property does not hold. Preliminary numerical experiments show that the presented penalty algorithm based on the coercive smoothed penalty gives very competitive results when compared with other penalty-based methods.