Blessings of maintaining infeasible solutions for constrained multi-objective optimization problems

Abstract

The most common approach to handling constraints in a constrained optimization problem has been the use of penalty functions. In recent years non-dominance based ranking methods have been applied for an efficient handling of constraints. These techniques favor the feasible solutions over the infeasible solutions, thus guiding the search through the feasible space. Usually the optimal solutions of the constrained optimization problems are spread along the constraint boundary. In this paper we propose a constraint handling method that maintains infeasible solutions in the population to aid the search of the optimal solutions through the infeasible space. The constraint handling method is implemented in constraint handling evolutionary algorithm (CHEA), which is the modified non-dominated sorting genetic algorithm II (NSGA-II) [1]. The original constrained minimization problem with k objectives is reformulated as an unconstrained minimization problem with k + 1 objectives, where an additional objective function is the number of constraint violations. In CHEA, the infeasible solutions are ranked higher than the feasible solutions, thereby focusing the search for the optimal solutions near the constraint boundaries through infeasible region. CHEA simultaneously obtains the solutions to the constrained as well as the unconstrained optimization problem. The performance of CHEA is compared with NSGA-II on the set of CTP test problems. For a fixed number of function evaluations, CHEA converges to the Pareto optimal solutions much faster than NSGA-II. It is observed that retaining even a small number of infeasible solutions in the population, CHEA is able to prevent the search from prematurely converging to a sub-optimal Pareto front.