Evolutionary Constrained Optimization: A Hybrid Approach

The holy grail of constrained optimization is the development of an efficient, scale invariant, and generic constraint-handling procedure in single- and multi-objective constrained optimization problems. Constrained optimization is a computationally difficult task, particularly if the constraint functions are nonlinear and nonconvex. As a generic classical approach, the penalty function approach is a popular methodology that degrades the objective function value by adding a penalty proportional to the constraint violation. However, the penalty function approach has been criticized for its sensitivity to the associated penalty parameters. Since its inception, evolutionary algorithms (EAs) have been modified in various ways to solve constrained optimization problems. Of them, the recent use of a bi-objective evolutionary algorithm in which the minimization of the constraint violation is included as an additional objective, has received significant attention. In this chapter, we propose a combination of a bi-objective evolutionary approach with the penalty function methodology in a manner complementary to each other. The bi-objective approach provides an appropriate estimate of the penalty parameter, while the solution of the unconstrained penalized function by a classical method induces a convergence property to the overall hybrid algorithm. We demonstrate the working of the procedure on a number of standard numerical test problems. In most cases, our proposed hybrid methodology is observed to take one or more orders of magnitude lesser number of function evaluations to find the constrained minimum solution accurately than some of the best-reported existing methodologies.