Numerical Methods For Vector Polynomial Optimization And A Class Of Distance Problem

Polynomial optimization is a class of important nonlinear programming,which has many significant practical applications.Lasserre semidefinite relaxation method is an important method for solving polynomial optimization problems,which has been widely studied and applied.Based on Lasserre semidefinite relaxation method,this paper discussed the numerical method of vector polynomial optimization and the distance between two closed semialgebraic sets.The first chapter briefly describes the research background of vector optimization,distance problem and polynomial optimization problems,and the main contents of this paper.The second chapter introduces Lasserre semidefinite relaxation method and other propaedeutics.In the third chapter,vector polynomial optimization problem is proposed.By using Lasserre semidefinite relaxation method in polynomial optimization,we study the main objective method,linear weighted sum method and ideal point method for solving vector polynomial optimization problem,respectively.Then,we proved that the solutions gotten by using these methods are weak efficient solutions(or efficient solutions)and numerical experiments showed the effictiveness of these methods.In chapter 4,numerical method for solving distance problem between two disjoint closed semialgebraic sets is studied.The problem is essentially a polynomial optimization problem.Lasserre semidefinite relaxation method can be applied to solve the shortest distance between the two sets.Numerical experiments indicate that the method is effective.The method can get approximative global optimal solutions,and does not depend on initial feasible solutions.Software package fmincon in Matlab does not have the two advantages.