This paper presents a new adaptive penalty method for genetic algorithms (GA). External penalty functions have been used to convert a constrained optimization problem into an unconstrained problem for GA-based optimization. The success of the genetic algorithm application to the design of water distribution systems depends on the choice of the penalty function. The optimal design of water distribution systems is a constrained non-linear optimization problem. Constraints (for example, the minimum pressure requirements at the nodes) are generally handled within genetic algorithm optimization by introducing a penalty cost function. The optimal solution is found when the pressures at some nodes are close to the minimum required pressure. The goal of an adaptive penalty function is to change the value of the penalty draw-down coefficient during the search allowing exploration of infeasible regions to find optimal building blocks, while preserving the feasibility of the final solution. In this study, a new penalty coefficient strategy is assumed to increase with the total cost at each generation and inversely with the total number of nodes. The application of the computer program to case studies shows that it finds the least cost in a favorable number of function evaluations if not less than that in previous studies and it is computationally much faster when compared with other studies.
Convergence of a distributed method for minimizing sum of convex functions with fixed point constraints
