The Pareto curve in an exact way for multi-objective differentiable functions with constraints

Abstract

Contrary to single objective functions where the calculation of optimal solutions will be done mathematically and accurately. On the other hand, multi-objective functions are more complicated to process in order to find the best solutions in an exact way. We start the exact resolution methods, because they make it possible to obtain a solution whose optimality is guaranteed, in certain situations, however, we can look for solutions of good quality, without guarantee of optimality, but in favor of a shorter calculation time. But they also show some signs of weakening, such as the number of objective functions to be optimized. For that reason a set of methods exists to study complicated systems. The objective is to find approached solutions. Their objective is to find the best possible high-quality solutions with reasonable calculation times. Many algorithms are used to solve this problem such as evolutionary algorithms, ant colony-based algorithms and swarm algorithms. But the major drawback of such algorithms is their precision and complexity. In this article, we propose a method which is based on a mathematical model to find the Pareto curve of multi objective functions of the functions derivable with constraints in an exact way. This method is based on the division of the principal problem into sub problems. Next, each sub-problem is studied using the gradient descent direction vector method in order to extract the solution ranges for each sub-problem. Then, use Pareto principle to extract the solutions building the Pareto curve for the principal problem. Finally, the proposed method is validated using the examples in the literature (BNH, SRN).