广义(多重集)分裂可行性问题的正则化算法研究

On the background of practical application for image reconstructions and the intensity modulated radiation therapy, the split feasibility problems become a research hotspot of nonlinear function analysis recently，We intend to research the iterative approximation solutions of the split feasibility problem and general split feasibility problems (split common fixed point problems、split variational inequality problems and split common null point problems)..First, we use the tool on fixed point of nonlinear operator and design the modified algorithms in order to analysis the convergence of the sequences. And the approximation theory of the solution of split feasibility problems will be studied. Secondly, the popular L1/2 regularization will be used to research the solution of split feasibility problems, the key technique is approximating L1/2 regularization by smoothing function to overcome its non-differentiable property, and the convergence theory are studied. Finally, since split common fixed point problems、split variational inequality problems and split common null point problems are recent new problems, lots of interested extension need to further studied, which mainly include designing new algorithms、expanding spaces and weaken restricted conditions and so on. This research can enrich and extend the split feasibility problems theory.

在图像处理与强度可调辐射疗法的实际应用背景下，分裂可行性问题成为近期非线性泛函分析的研究热点之一。 本项目拟从三个方面研究分裂可行性问题与广义分裂可行性问题（分裂公共不动点问题、分裂变分不等式问题和分裂公共零点问题）解的迭代逼近。.首先，利用非线性算子不动点理论工具，通过构造新的迭代算法，分析算法产生序列的收敛性，进而研究分裂可行性问题解的逼近理论。其次，采用最近流行的L1/2正则化方法改进求解分裂可行问题的算法，通过光滑函数逼近法克服其不可微性，研究其收敛性理论。最后，分裂公共不动点问题、分裂变分不等式问题和分裂公共零点问题是近期提出的新问题。因此，大量相关有意义的推广性问题亟待进一步研究，主要体现在新算法设计、空间扩展和参数减弱限制条件等方面。此研究有助于丰富和扩展分裂可行性问题理论。

在图像处理与强度可调辐射疗法的实际应用背景下，分裂可行性问题成为近期非线性泛函分析的研究热点之一。 本项目拟从三个方面研究分裂可行性问题与广义分裂可行性问题（分裂公共不动点问题、分裂变分不等式问题和分裂公共零点问题）解的迭代逼近。..首先，利用非线性算子不动点理论工具，通过构造新的迭代算法，分析算法产生序列的收敛性，进而研究分裂可行性问题解的逼近理论。其次，采用最近流行的L1/2正则化方法改进求解分裂可行问题的算法，通过光滑函数逼近法克服其不可微性，研究其收敛性理论。最后，分裂公共不动点问题、分裂变分不等式问题和分裂公共零点问题是近期提出的新问题。因此，大量相关有意义的推广性问题亟待进一步研究，主要体现在新算法设计、空间扩展和参数减弱限制条件等方面。此研究有助于丰富和扩展分裂可行性问题理论。