Semidefinite Programming Lagrangian Algorithm

Due to its extensive applications,semidefinite programming has become an active area of research and been in widespread use.In recent years,new theory and algorithms for semidefinite programming have been proposed.As nonlinear Lagrangian method is playing an important role in solving constrained optimization problems,this method is extended to solve nonconvex semidefinite programming.Firstly,this paper gives an introduction to semidefinite programming and analyzes optimality conditions for noncoverx semidefinite.Then,we analyze Lagrangian method for general nonconvex semidefinite programming,which is the algorithm based on new Lagrangian function.The main results obtained in this paper are concluded as follows:1.Chapter 1 reviews nonconvex semidefinite programming optimality conditions, which are the theory basis for the analysis of the Lagrangian algorithm for semidefinite programming.Based on previous theory,we establish the first order,second order optimality conditions of nonconvex semidefinite programming.2.Chapter2 analyzes a modified Lagrangian method for general nonconvex semidefinite programming.We analyze the characteristic of this Lagrangian function. Under some conditions,the local convergence of this method is also proved.Besides,the error bound of the solutions with the penalty parameter is established.It is proved that there exists a threshold of the penalty parameter such that the sequence products of the algorithm locally converge to the KKT point of the nonconvex semidefinite programming problem when the penalty parameter is less than the threshold.Some examples and preliminary computational results are also reported.3.In Chapter 3,Log-Sigmoid(LS)multipliers method is given,which use the Log-Sigmoid function to transform the semidefinite programming problem into a equivalent one.In the same way,we analyze the characteristic of the algorithm,and establish convergence of the LS multipliers method under very mild assumptions.Then we estimate the rate of convergence under the second order sufficient condition.