Smoothing Methods For Optimization Problem Based On L~? Minimization

It has a wide range of applications for solving the underdetermined system of linear equa-tions in the field of image restoration and reconstruction, and it has drawn increasing attention from scholars recent years. There are many important results about solving underdetermined linear system of equations problem, however, we don't see much deep study on methods for underdetermined system of linear equations based on l^? optimization problem. Therefore, this paper focuses on study of the smoothing methods of nonsmooth optimization based on l^? op-timization problem, namely the original problem is transformed to its smoothing approximate problem by smoothing l^? through the entropy function to solve.Firstly, the development and status quo of image reconstruction technology and the meth-ods of solving the nonsmooth optimization problems are briefly introduced. Then we introduce the smoothing method and show that the difference between the optimal values of the original function and the approximate one. We will research l^? minimization optimization problem where u ?Rn indicated the reconstructed image, A ?R^m×n indicated the sensor matrix, d ? R^m indicated the observed data.Generally, we translate l^? minimization optimization problem into l₂-l^? minimization optimization problem to solve, that is where A ? ?0,+??, we will convert the optimization problem into its smoothing approximation problem it can be solved by using the improved split Bregman iterative algorithm. In order to accel-erate the algorithm, we add the iterations of the smoothing parameter ? to Bregman iterative algorithm, and prove the convergence of the algorithm, that is, a sequence of solutions which is generated by the algorithm converges to the optimal solution of the original optimization problem in limited iterations. Finally, the numerical experiments show that the algorithm is effective, and we analyse the results of the experiment for different parameters.