Research Of Scaled Proximal Point Algorithms For Composite Optimization Problems

Many problems in the computer science,applied mathematics and management science come down to the composite optimization problem,which can be solved effectively by applying proximal point algorithms.In past decades,the research of this kind of algorithms has much progress,but the research on scaled proximal point algorithm is still rare.On the other hand,the superiorization method,a novel method proposed in recent years,has been shown to be useful in reducing the iteration steps or rich the results of original algorithms for solving problems such as computed tomography,compressed sensing and image reconstruction.Therefore,it has attracted the attention of many mathematicians and engineers.It is worth and necessary to find out more applications of this method.In this thesis,two scaled proximal point algorithms are proposed: proximal scaled gradient algorithm with multi-parameters and over-relaxed proximal scaled gradient algorithm.Two scaled proximal point algorithms and their corresponding inexact algorithms are proposed to solve composite optimization problem.The sequences generated by the algorithm and the inexact algorithm converge strongly to the solutions of the composite optimization problems under the assumption that the solution set of the problem is not empty.It is also proved that the proximal scaled gradient algorithms with multi-parameters have the property of bounded perturbation resilience.So that the superiorization algorithm is obtained.Finally,numerical examples have shown the effectiveness of the algorithms.The work of this thesis not only enrich the theoretical results of the proximal point algorithm,but also generalize the application practice of superiorization method.