Two Kinds Of Projection Algoirthms For Semidefinite-Programming

The research of Semidefinite programming problem was first started in1960s to1970s. As an extension of linear programming, Semidefinite programming is based on thecone, which is generated by the semidefinite matrix. Semidefinite programming is a classof convex optimization problem. Semidefinite programming has been widely applied inmany fields, such as, the problem of maximum eigenvalue optimization, the problem ofcombinatorial optimization, the robust optimization, systems and control theory, thestatistics and the structure optimization, etc. In order to solve the problem of semidefiniteprogramming in a better way, the researchers have put forward many effective and reliablealgorithms, such as, the interior point algorithm, the spectral bundle algorithm, theconjugate gradient method and the gradient method, etc. The researching and proposing ofthese algorithms have promoted the development of semidefinite programming.Semidefinite programming has become a very important research topic in the field ofoptimization now.At the beginning, the paper introduces the development and the present research stateof semidefinite programming, the basic theories, several related algorithms and the relatedknowledge and projection algorithms of variational inequality problems. Then, the papergive the KKT conditions for the existence of the optimal solution of the problem whensemidefinite programming satisfy the strictly feasible constraint conditions. By using theKKT conditions, semidefinite programming problem can be equivalently translateded intosolving variational inequality problems. At last, the paper present a new projection classalgorithm and the improved algorithm for solving semidefinite programming problem,which borrows from the projection algorithm of variational inequality problem. This papermainly accomplished the following two parts of work:1. At First, We present a new projection class algorithm for solving semidefiniteprogramming problem, which reference from the projection algorithm of variationalinequality problem. Then, we prove that the algorithm has the global convergence whenthe optimal solution set of the problem is not empty and the function F ispseudo-monotone. Finally, the researcher give the results of numerical experiment of newalgorithms for the test problem. The results of numerical experiment show that the algorithm is feasible.2. As to the first projection algorithm, the author gives an improved algorithm andproves that the improved algorithm also has the global convergence in the sameassumption conditions. From the analysis and the proving of the improved algorithm, weknow that the iterative pointuk1of the improved algorithm is closer to the optimumsolution u than the first algorithm.