哈尔滨工业大学理学硕士学位论文
Abstract
The sparse optimization problems have a wide variety of applications such as signal processing,image denoising and visual coding.The main goal of these problems is to find a solution in which most of its elements are zeros.Therefore,the optimization model with cardinality terms is the most direct and ideal model for solving sparse optimization problems.
In this thesis,we first propose an accelerated projection gradient algorithm for solving the constrained convex optimization problems.We show that the algorithm has the exponential worst case computational complexity when the objective function is strongly convex.
We then propose an accelerated IHT algorithm for solving the
l regularized convex
0 optimization problem with box constraints.We first substantiate that there exists a threshold,such that if the extrapolation coefficients are chosen below this threshold,the proposed algorithm is equivalent to the accelerated projected gradient algorithm for solving a convex optimization problem after finite iterations.Under the error bound condition of the data fitting function,we prove the iterate sequence is R-linearly convergent to a local minimizer of the problem.Moreover,the corresponding sequence of objective function values is also R-linearly convergent.
Keywords:sparse optimization problem,accelerated IHT algorithm,
l regularization,
linear convergence
哈尔滨工业大学理学硕士学位论文
目录
摘要................................................................................................................................I Abstract.............................................................................................................................II 第1章绪论.. (1)
1.1课题来源及研究的背景和意义 (1)
1.2国内外研究现状及分析 (1)
1.3主要研究内容 (3)
第2章预备知识 (5)
2.1涉及的符号 (5)
2.2涉及的定义及性质 (5)
2.3本章小结 (7)
第3章约束凸规划问题 (8)
3.1引言 (8)
3.2算法I (8)
3.3算法I的收敛性 (9)
3.4算法II (13)
3.5算法II的收敛性 (14)
3.6本章小结 (19)
l正则化凸规划 (20)
第4章盒约束
4.1引言 (20)
4.2算法III (20)
4.3算法III的收敛性 (21)
4.4数值实验 (28)
4.5本章小结 (31)
结论 (33)
参考文献 (34)
哈尔滨工业大学学位论文原创性声明及使用授权说明 (38)
致谢 (39)