求解非光滑、非凸正则极小化问题的光滑化信赖域方法

Recently, nonsmooth,nonconvex optimizations has attracted significant attention in engineering and economics. An increasing number of practical problems require solving the nonsmooth,nonconvex optimization problems. Smoothing approximations for optimization problems have been studied for decades and become an important tool for solving nonsmooth minimization. Line search and trust region are two major strategies for continuous optimization. Trust region methods for solving nonsmooth optimization problems have been studied for long time as well. However, there is a little attention on combining smoothing approximations and trust region methods. Most of exist smoothing methods are in the framework of line search and only seldom works about smoothing trust region can be found. This project is focus on smoothing trust region methods for the regularized minimization problems with nonconvex,nonsmooth, perhaps non-Lipschitz penalty functions,which attracted considerable attention in many applications including image restoration, signal reconstruction, variable selection. The main work will include: (1) Derive the first order and second order necessary optimality conditions and sufficient optimality conditions for local minimizers of such minimization problems. (2) Construct the effective smoothing approximations for the specific penalty functions. (3) Design the global convergent smoothing trust region method which can find a point satisfying the first order or second order necessary optimality conditions from any starting point.(4) Analyze the local convergent rate of iterates for the locally Lipschitz continuous case and the computational complexity of the algorithms for the non-Lipschitz case. (5) Apply new methods to practical applications. For most part of our research contents in this project, some are totally new topics, some are still in their infancy. Besides provide new methods for a class nonsmooth, nonconvex minimizations, the study of this project can also enrich the theory and techniques in trust region and smoothing methods.

非凸、非光滑优化问题在工程和经济中备受关注。光滑逼近作为处理函数非光滑性的主要手段，已成为求解非光滑优化的重要工具。在现有的光滑化方法中，和大多数采用线搜索策略的情形相比，有关光滑化信赖域方法的研究并不多见。因此，本项目计划研究如何利用光滑化信赖域方法求解一类带有非凸、非光滑罚函数项的正则极小化问题。这类问题近年来在图像恢复、信号重构、变量选择等众多领域有广泛的应用。项目将着重研究： 1）探讨怎样刻画所研究问题的最优性条件；（2）研究如何针对罚函数的结构特点构造有效的光滑逼近函数；（3）设计全局收敛的光滑化信赖域方法；（4）分析算法的局部收敛速度（对局部Lipschitz连续情形）或计算复杂度（对非局部Lipschitz连续情形）；（5）将新方法应用于实际。本项目的开展不仅为一类具有广泛应用价值的非光滑问题提供新的求解方法，也能进一步发展和丰富信赖域和光滑化方法本身的理论和技术。

非凸、非光滑优化问题在工程和经济中备受关注。光滑逼近作为处理函数非光滑性的主要手段，已成为求解非光滑最优化的重要工具。在现有的光滑化方法中，和大多数采用线搜索策略的情形相比，有关光滑化信赖域方法的研究并不多见。因此，本项目研究了如何利用光滑化信赖域方法求解一类带有非凸、非光滑罚函数项的正则极小化问题。这类问题近年来在图像恢复、信号重构、变量选择等众多领域有广泛的应用。项目着重研究了一下几方面的内容： 1）探讨怎样刻画所研究问题的最优性条件；（2）研究如何针对罚函数的结构特点构造有效的光滑逼近函数；（3）设计全局收敛的光滑化信赖域方法；（4）分析算法的局部收敛速度（对局部Lipschitz 连续情形）或计算复杂度（对非局部Lipschitz 连续情形）；（5）将新方法应用于实际。本项目的开展不仅为一类具有广泛应用价值的非光滑问题提供新的求解方法，也进一步发展和丰富了信赖域和光滑化方法本身的理论和技术。