The Study On Projection Methods For Semidefinite Programming

Semidefinite programming is an extension of linear programming. In semidefinite programming one maximizes (minimizes) a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programming is a nonsmooth convex optimization problem. In recent years, semidefinite programming has been one of the most active research areas in mathematical programming because its theory and algorithms have developed greatly and its numerous applications have been found in control theory, electrical engineering and combinatorial optimization.In this paper, we firstly summarize the theory, algorithms, applications and research state of semidefinite programming, secondly introduce ours some work in algorithms for semidefinite programming. For detail, we conclude them as follows:1.We first transform a semidefinite programming into a variational inequality problem, and then propose a novel prediction-correction algorithm based on Korpelevich-Khobotv method under monotone and Lipschitz continuous conditions. The convergence of the algorithm is also given and the optimal stepsize is employed in the correction step to accelerate convergence rate. The numerical experiment indicates that this method is effective.2.Semidefinite.programming is approximated via constructing a series of strongly monotone variational inequality problems with same structure. A new iterative algorithm for semidefinite programming is obtained by solving these strongly monotone variational inequality problems and is applied to the Max-bisection problem. Numerical results demonstrate that the algorithm is very effective for Max-bisection problem.