Improved Multi-objective Heterogeneous Trust Region Algorithm

The multi-objective optimization problem is an optimization problem in which two or more objective functions are obtained at the same time.In multi-objective optimization problems,the objective function is often heterogeneous,that is,the function value of one objective function can only be obtained through a large number of calculations,and the derivative cannot be obtained within a reasonable calculation cost.For other objective functions,analytical expressions are given,whose function values and derivatives are easy to obtain.Usually the goal of a multi-objective optimization problem is to approximate the entire effective point set,but if the calculation of the objective function is very large,it becomes impractical to approximate the entire effective point set.In response to this problem,this paper proposes an improved multi-objective heterogeneous trust region algorithm(MHSGT),the specific work is as follows:First,change the update method of the interpolation point set in the multi-objective heterogeneous trust region algorithm(MHT).The MHT algorithm uses the maximized Lagrangian polynomial function to improve the balance of the interpolation point set.The method does not consider the distance between the points.Therefore,this paper combines the simple geometric improvement method with the self-correcting geometric method,and proposes an improved self-correcting geometric method.This method ensures that the interpolation point set has good geometric properties,thereby improving the approximation effect of the model function.Second,change the acceptance ratio of test points in the MHT algorithm.In this paper,the acceptance ratio of test points in the single-object trust region method is directly extended to the multi-objective situation.The new ratio ensures that each objective function has a sufficient amount of decline in each successful iteration,reducing the amount of unnecessary function calculations.Finally,the improved algorithm,MHSGT algorithm,is analyzed for convergence,which proves that the improved algorithm still converges to the Pareto critical point of the original problem.At the same time,through the results of numerical experiments,it is known that overall,the calculation amount generated by the MHT algorithm and the MHSGT algorithm is not much different;but for overlapping examples,the number of iterations required by the MHSGT algorithm is repeated,and the amount of calculation is smaller.