Hybrid Algorithm And Testing Algorithm Of Proper Efficient Solution For Vector Polynomial Optimization Problem

Vector optimization problem is an important mathematical programming problem,which appears widely in practical problems.Vector polynomial optimization problem is a kind of common vector optimization problem with polynomial structure.It is important to study its numerical algorithm.In this paper,a hybrid algorithm is designed for the vector polynomial optimization problem and a numerical algorithm is designed to check whether the efficient solution obtained by solving the problem is a proper efficient solution.Chapter 1 describes the research background of vector optimization problem,polynomial optimization problem and the main content of this paper.Chapter 2 introduces the preliminary knowledge of polynomial optimization,Lasserre semi-positive definite relaxation methods,coercive polynomials and others.In Chapter 3,we propose a hybrid algorithm for vector polynomial optimization problem when feasible set is a compact set.Firstly,the hybrid method is combined with Lasserre semi-positive definite relaxation method to solve the problem.Secondly,the relationship between the efficient solution of the original problem and the optimal solution of the transformed problem is analyzed.Thirdly,we analyze the convergence of the algorithm.Finally,numerical experiments show that the algorithm is effective.In Chapter 4,we propose a hybrid algorithm for vector polynomial optimization problem when feasible set is not a compact set.Firstly,based on the hybrid method,the feasible set is transformed into a compact set by using the extended quadratic module.Secondly,the problem is transformed into a semi-positive definite programming problem to solve.Thirdly,we analyze the convergence of the algorithm.Finally,numerical experiments show that the algorithm is effective.Chapter 5 presents an algorithm to check whether the efficient solution is a proper efficient solution or not.Firstly,we design an algorithm according to the definition of the proper efficient solution.Secondly,ideal point method and linear scalarization method are used to solve vector polynomial optimization problems respectively.Thirdly,the validity of the algorithm is verified by numerical experiments.Finally,the numerical results are consistent with the conclusions of relevant theorems.