Lagrange Duality Theory Of Semidefinite Programming And Its Application In Location Problem

Semidefinite programming is widely existed in many fields,such as system and control theory,financial engineering,quantum chemistry,signal processing,and so on.The dual theory plays a very important role in the theoretical research and algorithm design of the optimization problem.This paper focuses on Lagrange duality theory of semidefinite programming and its application in location problem.To specific:1.Chapter 2,it is concentrated on Lagrange duality theory of semidefinite program-ming.Firstly,the discretization method of the primal semidefinite programming problem and the dual semidefinite programming problem is given in this paper.The dual semidef-inite programming is then converted into a linear programming problem approximately via the method.Secondly,based on the discretization method and its convergence prop-erty,we prove the Lagrangian strong duality theorem in the case of slater conditions satisfied for the primal semidefinite programming problem,and both the primal and the dual semidefinite programming problems.Finally,the proposed new proof provides a brand new algorithm to solve the semidefinite programs via semiinfinite discretization methods in theory and its convergence analysis is also considered.2.Chapter 3,it is focus on the application of Lagrange duality theory to the semidefi-nite programming in the location problem.Firstly,a class of minimax location problem is considered in this paper.Then,given a new semidefinite programming relaxation method for the location problem.Finally,based on the Lagrangian strong duality theorem of the semidefinite programming problem and the Gershgorin circle thereorem we prove the strong dual results of the semidefinite programming relaxation problem and its Lagrangian dual problem.