Interactive Newton's Method For Multi-objective Optimization With Constraints

We propose an interactive Newton's method for multi-objective optimization withconstraints by presenting the notation of generalized critical points and creating theNewton's direction. We need not scalarize the original multi-objective optimizationproblem, and neither ordering information nor weighting factors for the objective func-tions are needed be given by the decision maker.First we propose a generalized Newton's method for multi-objective program-ming. Under the conditions that the objective functions and constraint functions areassumed to be twice continuously differentiable, we create the Newton's directionwhich is a feasible direction and is a descent direction for every objective functions.Finally, by using the Armijo step along this direction, an approximate generalizedcritical point can be obtained step by step.Then an interactive Newton's method is given on the basis of the generalizedNewton's method. The procedure of the interactive method can be divided into twostages, the analyzing stage and the decision-making stage respectively. In the formerstage, we get an approximate generalized critical point by the generalized Newton'smethod. In the latter stage, the decision maker should give a classification of theobjective functions according to his preference and the allowed amount of increase tosome objective functions whose values have been accepted. By this way, we can get anew multi-objective optimization with constraints and go to the analyzing stage againusing the approximate generalized critical point obtained in the former stage as theinitial point.In this paper, the convergence result of this Newton's method as well as the nu-merical experiment with good results is provided.