Hard-thresholding-based algorithms have seen various advantages for sparse optimization in controlling the sparsity and allowing for fast computation. Recent research shows that when techniques of the Newton-type methods are integrated, their numerical performance can be improved surprisingly. This paper develops a gradient projection Newton pursuit algorithm that mainly adopts the hard-thresholding operator and employs the Newton pursuit only when certain conditions are satisfied. The proposed algorithm is capable of converging globally and quadratically under the standard assumptions. When it comes to compressive sensing problems, the imposed assumptions are much weaker than those for many state-of-the-art algorithms. Moreover, extensive numerical experiments have demonstrated its high performance in comparison with the other leading solvers.
A Hybrid Quasi-Newton Projected-Gradient Method with Application to Lasso and Basis-Pursuit Denoise
本文提出了一种新的算法，用于Rn中多面体集上凸函数的优化。该算法扩展了谱投影梯度法，尽可能将有限内存BFGS迭代限制在当前人脸上。我们在适当的条件下证明了算法的收敛性，并将该算法应用于解决Lasso问题，从而通过van den Berg和Friedlander [SIAM Journal on Scientific Computing，31（2），2008]提出的寻根框架解决了基追踪去噪问题。该算法特别适用于简单区域，也可用于解决有界约束问题以及限制在单纯形的问题。
Efficient Relaxed Gradient Support Pursuit for Sparsity Constrained Non-convex Optimization
Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education（教育部智能感知与图像理解重点实验室）；School of Artificial Intelligence, Xidian University（西安电子科技大学人工智能学院）；Department of Computer Science, The University of Texas at Austin（德克萨斯大学奥斯汀分校计算机科学系）
Preconditioning PDE-constrained optimization with $\rm L^1$-sparsity and control constraints
Università degli Studi di Firenze（佛罗伦萨大学）；Università di Bologna（博洛尼亚大学）；Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）
